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• Problems in Mathematics
• Elementary Number Theory
  • Even Perfect Numbers and Mersenne Prime Numbers
• Field Theory
  • The Number of Elements in a Finite Field is a Power of a Prime Number
  • Prove that $\F_3[x]/(x^2+1)$ is a Field and Find the Inverse Elements
  • Each Element in a Finite Field is the Sum of Two Squares
  • Any Automorphism of the Field of Real Numbers Must be the Identity Map
  • Example of an Infinite Algebraic Extension
  • The Cyclotomic Field of 8-th Roots of Unity is $\Q(\zeta_8)=\Q(i, \sqrt{2})$
  • A Rational Root of a Monic Polynomial with Integer Coefficients is an Integer
  • Cubic Polynomial $x^3-2$ is Irreducible Over the Field $\Q(i)$
  • Prove that any Algebraic Closed Field is Infinite
  • Extension Degree of Maximal Real Subfield of Cyclotomic Field
  • Equation $x_1^2+\cdots +x_k^2=-1$ Doesn't Have a Solution in Number Field $\Q(\sqrt[3]{2}e^{2\pi i/3})$
  • Application of Field Extension to Linear Combination
  • Irreducible Polynomial $x^3+9x+6$ and Inverse Element in Field Extension
  • Explicit Field Isomorphism of Finite Fields
  • Galois Extension $\Q(\sqrt{2+\sqrt{2}})$ of Degree 4 with Cyclic Group
  • Galois Group of the Polynomial $x^2-2$
  • Polynomial $x^p-x+a$ is Irreducible and Separable Over a Finite Field
  • Show that Two Fields are Equal: $\Q(\sqrt{2}, \sqrt{3})= \Q(\sqrt{2}+\sqrt{3})$
  • Galois Group of the Polynomial $x^p-2$.
  • Two Quadratic Fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are Not Isomorphic
  • Automorphism Group of $\Q(\sqrt[3]{2})$ Over $\Q$.
  • Determine the Splitting Field of the Polynomial $x^4+x^2+1$ over $\Q$
  • In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.
  • The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity
  • Algebraic Number is an Eigenvalue of Matrix with Rational Entries
  • Degree of an Irreducible Factor of a Composition of Polynomials
  • $x^3-\sqrt{2}$ is Irreducible Over the Field $\Q(\sqrt{2})$
• General
  • Mathematics About the Number 2018
  • Top 10 Popular Math Problems in 2016-2017
  • Even Perfect Numbers and Mersenne Prime Numbers
  • A One-Line Proof that there are Infinitely Many Prime Numbers
  • Beautiful Formulas for pi=3.14...
  • Mathematics About the Number 2017
  • Find the Largest Prime Number Less than One Million.
• Group Theory
  • Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57
  • If There are 28 Elements of Order 5, How Many Subgroups of Order 5?
  • Union of Two Subgroups is Not a Group
  • Normal Subgroup Whose Order is Relatively Prime to Its Index
  • Every Cyclic Group is Abelian
  • The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$
  • The Number of Elements Satisfying $g^5=e$ in a Finite Group is Odd
  • Group Homomorphism from $\Z/n\Z$ to $\Z/m\Z$ When $m$ Divides $n$
  • Example of an Infinite Group Whose Elements Have Finite Orders
  • If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order
  • Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8
  • Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4
  • If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$
  • If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial
  • The Product of Distinct Sylow $p$-Subgroups Can Never be a Subgroup
  • The Normalizer of a Proper Subgroup of a Nilpotent Group is Strictly Bigger
  • Elements of Finite Order of an Abelian Group form a Subgroup
  • The Additive Group of Rational Numbers and The Multiplicative Group of Positive Rational Numbers are Not Isomorphic
  • The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements
  • Every Finite Group Having More than Two Elements Has a Nontrivial Automorphism
  • If Two Subsets $A, B$ of a Finite Group $G$ are Large Enough, then $G=AB$
  • A Group Homomorphism that Factors though Another Group
  • If a Finite Group Acts on a Set Freely and Transitively, then the Numbers of Elements are the Same
  • Every Group of Order 72 is Not a Simple Group
  • A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal
  • If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup
  • Example of Two Groups and a Subgroup of the Direct Product that is Not of the Form of Direct Product
  • The Symmetric Group is a Semi-Direct Product of the Alternating Group and a Subgroup $\langle(1,2) \rangle$
  • Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$
  • Every Group of Order 20449 is an Abelian Group
  • The Group of Rational Numbers is Not Finitely Generated
  • Every Finitely Generated Subgroup of Additive Group $\Q$ of Rational Numbers is Cyclic
  • Prove that a Group of Order 217 is Cyclic and Find the Number of Generators
  • The Order of a Conjugacy Class Divides the Order of the Group
  • The Product of a Subgroup and a Normal Subgroup is a Subgroup
  • Inverse Map of a Bijective Homomorphism is a Group Homomorphism
  • Group Homomorphism Sends the Inverse Element to the Inverse Element
  • Injective Group Homomorphism that does not have Inverse Homomorphism
  • Fundamental Theorem of Finitely Generated Abelian Groups and its application
  • Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $
  • Prove a Group is Abelian if $(ab)^2=a^2b^2$
  • $p$-Group Acting on a Finite Set and the Number of Fixed Points
  • Order of Product of Two Elements in a Group
  • Group Homomorphisms From Group of Order 21 to Group of Order 49
  • Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies ^{n+1}|p-1$.
  • Abelian Normal subgroup, Quotient Group, and Automorphism Group
  • Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups
  • If Quotient $G/H$ is Abelian Group and $H < K \triangleleft　G$, then $G/K$ is Abelian
  • Quotient Group of Abelian Group is Abelian
  • Special Linear Group is a Normal Subgroup of General Linear Group
  • If the Order of a Group is Even, then the Number of Elements of Order 2 is Odd
  • A Group is Abelian if and only if Squaring is a Group Homomorphism
  • The Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent Function
  • Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian Group
  • If a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian Group
  • Non-Abelian Group of Order $pq$ and its Sylow Subgroups
  • The Order of $ab$ and $ba$ in a Group are the Same
  • A Simple Abelian Group if and only if the Order is a Prime Number
  • A Group of Order $ is Solvable
  • The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$
  • Sylow Subgroups of a Group of Order 33 is Normal Subgroups
  • Eckmann–Hilton Argument: Group Operation is a Group Homomorphism
  • Equivalent Definitions of Characteristic Subgroups. Center is Characteristic.
  • Group of Order $pq$ Has a Normal Sylow Subgroup and Solvable
  • Pullback Group of Two Group Homomorphisms into a Group
  • A Group Homomorphism is Injective if and only if Monic
  • No Finite Abelian Group is Divisible
  • Subgroup of Finite Index Contains a Normal Subgroup of Finite Index
  • Subgroup Containing All $p$-Sylow Subgroups of a Group
  • If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup
  • Cyclic Group if and only if There Exists a Surjective Group Homomorphism From $\Z$
  • Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself
  • Use Lagrange's Theorem to Prove Fermat's Little Theorem
  • If Every Nonidentity Element of a Group has Order 2, then it's an Abelian Group
  • Group Homomorphism, Conjugate, Center, and Abelian group
  • Group Homomorphism, Preimage, and Product of Groups
  • A Group Homomorphism and an Abelian Group
  • Order of the Product of Two Elements in an Abelian Group
  • Two Normal Subgroups Intersecting Trivially Commute Each Other
  • Abelian Normal Subgroup, Intersection, and Product of Groups
  • Abelian Groups and Surjective Group Homomorphism
  • A Homomorphism from the Additive Group of Integers to Itself
  • Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup
  • Finite Group and Subgroup Criteria
  • Non-Abelian Simple Group is Equal to its Commutator Subgroup
  • Two Quotients Groups are Abelian then Intersection Quotient is Abelian
  • Commutator Subgroup and Abelian Quotient Group
  • Finite Group and a Unique Solution of an Equation
  • A Group Homomorphism is Injective if and only if the Kernel is Trivial
  • Multiplicative Groups of Real Numbers and Complex Numbers are not Isomorphic
  • Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup
  • If a Subgroup $H$ is in the Center of a Group $G$ and $G/H$ is Nilpotent, then $G$ is Nilpotent
  • Normal Subgroups, Isomorphic Quotients, But Not Isomorphic
  • The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$
  • Infinite Cyclic Groups Do Not Have Composition Series
  • Any Finite Group Has a Composition Series
  • Group of Order 18 is Solvable
  • If a Subgroup Contains a Sylow Subgroup, then the Normalizer is the Subgroup itself
  • The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal
  • Isomorphism Criterion of Semidirect Product of Groups
  • Nontrivial Action of a Simple Group on a Finite Set
  • Conjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the Set
  • Group of Invertible Matrices Over a Finite Field and its Stabilizer
  • If a Group is of Odd Order, then Any Nonidentity Element is Not Conjugate to its Inverse
  • A Subgroup of the Smallest Prime Divisor Index of a Group is Normal
  • Are Groups of Order 100, 200 Simple?
  • Abelian Group and Direct Product of Its Subgroups
  • Normalizer and Centralizer of a Subgroup of Order 2
  • A Group of Order $pqr$ Contains a Normal Subgroup of Order Either $p, q$, or $r$
  • If the Order is an Even Perfect Number, then a Group is not Simple
  • Sylow's Theorem (Summary)
  • All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8
  • Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$
  • Dihedral Group and Rotation of the Plane
  • Normal Subgroups Intersecting Trivially Commute in a Group
  • The Center of the Symmetric group is Trivial if $n>2$
  • Group of Order $pq$ is Either Abelian or the Center is Trivial
  • Basic Properties of Characteristic Groups
  • A Group of Order the Square of a Prime is Abelian
  • If the Quotient by the Center is Cyclic, then the Group is Abelian
  • A Group with a Prime Power Order Elements Has Order a Power of the Prime.
  • Any Subgroup of Index 2 in a Finite Group is Normal
  • The Center of a p-Group is Not Trivial
  • A Group of Linear Functions
  • The Quotient by the Kernel Induces an Injective Homomorphism
  • A Condition that a Commutator Group is a Normal Subgroup
• Linear Algebra
  • If the Nullity of a Linear Transformation is Zero, then Linearly Independent Vectors are Mapped to Linearly Independent Vectors
  • Find All Values of $x$ such that the Matrix is Invertible
  • Find All Eigenvalues and Corresponding Eigenvectors for the \times 3$ matrix
  • Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3.
  • Compute the Determinant of a Magic Square
  • Are These Linear Transformations?
  • Using Gram-Schmidt Orthogonalization, Find an Orthogonal Basis for the Span
  • Normalize Lengths to Obtain an Orthonormal Basis
  • Find a Spanning Set for the Vector Space of Skew-Symmetric Matrices
  • Determine Bases for Nullspaces $\calN(A)$ and $\calN(A^{T}A)$
  • In which $\R^k$, are the Nullspace and Range Subspaces?
  • Prove Vector Space Properties Using Vector Space Axioms
  • Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors
  • Find a Basis for the Subspace spanned by Five Vectors
  • How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix
  • Can We Reduce the Number of Vectors in a Spanning Set?
  • Does an Extra Vector Change the Span?
  • Vector Space of Functions from a Set to a Vector Space
  • Find a Basis for Nullspace, Row Space, and Range of a Matrix
  • Describe the Range of the Matrix Using the Definition of the Range
  • True or False Problems on Midterm Exam 1 at OSU Spring 2018
  • Find the Vector Form Solution to the Matrix Equation $A\mathbf{x}=\mathbf{0}$
  • If $\mathbf{v}, \mathbf{w}$ are Linearly Independent Vectors and $A$ is Nonsingular, then $A\mathbf{v}, A\mathbf{w}$ are Linearly Independent
  • Find a Nonsingular Matrix $A$ satisfying A=A^2+AB$
  • Determine whether the Matrix is Nonsingular from the Given Relation
  • Find All Symmetric Matrices satisfying the Equation
  • Compute $A^5\mathbf{u}$ Using Linear Combination
  • If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent?
  • Using Properties of Inverse Matrices, Simplify the Expression
  • Elementary Questions about a Matrix
  • Are these vectors in the Nullspace of the Matrix?
  • Spanning Sets for $\R^2$ or its Subspaces
  • Is the Derivative Linear Transformation Diagonalizable?
  • Dot Product, Lengths, and Distances of Complex Vectors
  • How to Obtain Information of a Vector if Information of Other Vectors are Given
  • Inner Products, Lengths, and Distances of 3-Dimensional Real Vectors
  • Given the Data of Eigenvalues, Determine if the Matrix is Invertible
  • A Recursive Relationship for a Power of a Matrix
  • The Rotation Matrix is an Orthogonal Transformation
  • The Coordinate Vector for a Polynomial with respect to the Given Basis
  • Find a Basis for the Range of a Linear Transformation of Vector Spaces of Matrices
  • The Matrix Exponential of a Diagonal Matrix
  • Find the Nullspace and Range of the Linear Transformation $T(f)(x) = f(x)-f(0)$
  • Find the Matrix Representation of $T(f)(x) = f(x^2)$ if it is a Linear Transformation
  • Is the Map $T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3$ a Linear Transformation?
  • Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions?
  • The Rank and Nullity of a Linear Transformation from Vector Spaces of Matrices to Polynomials
  • Taking the Third Order Taylor Polynomial is a Linear Transformation
  • Is the Map $T (f) (x) = f(x) - x - 1$ a Linear Transformation between Vector Spaces of Polynomials?
  • The Matrix Representation of the Linear Transformation $T (f) (x) = ( x^2 - 2) f(x)$
  • The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$
  • Determine whether the Given 3 by 3 Matrices are Nonsingular
  • For What Values of $a$, Is the Matrix Nonsingular?
  • Are Coefficient Matrices of the Systems of Linear Equations Nonsingular?
  • Solving a System of Differential Equation by Finding Eigenvalues and Eigenvectors
  • Solve the Linear Dynamical System $\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} =A\mathbf{x}$ by Diagonalization
  • Prove that $\{ 1 , 1 + x , (1 + x)^2 \}$ is a Basis for the Vector Space of Polynomials of Degree $ or Less
  • For Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace
  • A Line is a Subspace if and only if its $y$-Intercept is Zero
  • Determine the Values of $a$ so that $W_a$ is a Subspace
  • The Set $ \{ a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$ is a Subspace in $C(\R)$
  • The Centralizer of a Matrix is a Subspace
  • The Set of Vectors Perpendicular to a Given Vector is a Subspace
  • Prove that the Center of Matrices is a Subspace
  • If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$
  • A Condition that a Vector is a Linear Combination of Columns Vectors of a Matrix
  • Show that the Given 2 by 2 Matrix is Singular
  • Column Vectors of an Upper Triangular Matrix with Nonzero Diagonal Entries are Linearly Independent
  • Write a Vector as a Linear Combination of Three Vectors
  • Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent
  • Determine Trigonometric Functions with Given Conditions
  • Find a Quadratic Function Satisfying Conditions on Derivatives
  • Determine a 2-Digit Number Satisfying Two Conditions
  • Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems
  • If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal?
  • Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2
  • If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix
  • If Two Matrices Have the Same Rank, Are They Row-Equivalent?
  • Find a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the Rank
  • Row Equivalence of Matrices is Transitive
  • Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$
  • Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$
  • The Length of a Vector is Zero if and only if the Vector is the Zero Vector
  • A Relation between the Dot Product and the Trace
  • Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$
  • Matrix Operations with Transpose
  • Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components
  • Does the Trace Commute with Matrix Multiplication? Is $\tr (A B) = \tr (A) \tr (B) $?
  • Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix?
  • The Vector $S^{-1}\mathbf{v}$ is the Coordinate Vector of $\mathbf{v}$
  • Express the Eigenvalues of a 2 by 2 Matrix in Terms of the Trace and Determinant
  • Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix
  • Diagonalize a 2 by 2 Symmetric Matrix
  • Is the Following Function $T:\R^2 \to \R^3$ a Linear Transformation?
  • If the Sum of Entries in Each Row of a Matrix is Zero, then the Matrix is Singular
  • Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$
  • Is the Set of All Orthogonal Matrices a Vector Space?
  • Linear Transformation $T:\R^2 \to \R^2$ Given in Figure
  • Eigenvalues of \times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials
  • If Matrices Commute $AB=BA$, then They Share a Common Eigenvector
  • Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less
  • Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors
  • Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$
  • Find Bases for the Null Space, Range, and the Row Space of a \times 4$ Matrix
  • Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?
  • Find an Orthonormal Basis of the Given Two Dimensional Vector Space
  • Vector Space of 2 by 2 Traceless Matrices
  • Find an Orthonormal Basis of $\R^3$ Containing a Given Vector
  • The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive
  • Linear Transformation that Maps Each Vector to Its Reflection with Respect to $x$-Axis
  • An Example of a Real Matrix that Does Not Have Real Eigenvalues
  • The Intersection of Two Subspaces is also a Subspace
  • Eigenvalues and Eigenvectors of The Cross Product Linear Transformation
  • An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism
  • Orthogonal Nonzero Vectors Are Linearly Independent
  • Exponential Functions Form a Basis of a Vector Space
  • Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less
  • Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues
  • The set of \times 2$ Symmetric Matrices is a Subspace
  • Diagonalize the \times 2$ Hermitian Matrix by a Unitary Matrix
  • A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix
  • Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix
  • Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible?
  • The Subspace of Linear Combinations whose Sums of Coefficients are zero
  • The Column Vectors of Every \times 5$ Matrix Are Linearly Dependent
  • Determine Whether Each Set is a Basis for $\R^3$
  • Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors
  • Every Basis of a Subspace Has the Same Number of Vectors
  • If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent
  • Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.
  • Linear Algebra Midterm 1 at the Ohio State University (3/3)
  • Linear Algebra Midterm 1 at the Ohio State University (2/3)
  • Linear Algebra Midterm 1 at the Ohio State University (1/3)
  • An Example of Matrices $A$, $B$ such that $\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B)$
  • The Powers of the Matrix with Cosine and Sine Functions
  • An Example of a Matrix that Cannot Be a Commutator
  • 7 Problems on Skew-Symmetric Matrices
  • Determine a Condition on $a, b$ so that Vectors are Linearly Dependent
  • Two Matrices are Nonsingular if and only if the Product is Nonsingular
  • A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors
  • The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$.
  • Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$
  • The Transpose of a Nonsingular Matrix is Nonsingular
  • Construction of a Symmetric Matrix whose Inverse Matrix is Itself
  • The Range and Null Space of the Zero Transformation of Vector Spaces
  • Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism
  • Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
  • The Sum of Cosine Squared in an Inner Product Space
  • Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors
  • Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent
  • A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$
  • Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$
  • Find Inverse Matrices Using Adjoint Matrices
  • Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$
  • A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$
  • A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero
  • The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization
  • A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space
  • True or False: If $A, B$ are 2 by 2 Matrices such that $(AB)^2=O$, then $(BA)^2=O$
  • Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$
  • Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism?
  • Unit Vectors and Idempotent Matrices
  • A Positive Definite Matrix Has a Unique Positive Definite Square Root
  • Find All the Square Roots of a Given 2 by 2 Matrix
  • No/Infinitely Many Square Roots of 2 by 2 Matrices
  • How to Prove a Matrix is Nonsingular in 10 Seconds
  • Eigenvalues of a Matrix and its Transpose are the Same
  • The Inverse Matrix of the Transpose is the Transpose of the Inverse Matrix
  • The Formula for the Inverse Matrix of $I+A$ for a \times 2$ Singular Matrix $A$
  • Every Diagonalizable Nilpotent Matrix is the Zero Matrix
  • How to Use the Cayley-Hamilton Theorem to Find the Inverse Matrix
  • 10 True of False Problems about Nonsingular / Invertible Matrices
  • The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane
  • A Matrix Commuting With a Diagonal Matrix with Distinct Entries is Diagonal
  • Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$
  • Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
  • Given the Characteristic Polynomial, Find the Rank of the Matrix
  • Diagonalize the 3 by 3 Matrix Whose Entries are All One
  • Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable
  • Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  • Determine Whether Given Subsets in $\R^4$ are Subspaces or Not
  • The Product of Two Nonsingular Matrices is Nonsingular
  • Find an Orthonormal Basis of the Range of a Linear Transformation
  • Diagonalize a 2 by 2 Matrix if Diagonalizable
  • Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
  • Find All the Eigenvalues of 4 by 4 Matrix
  • The Determinant of a Skew-Symmetric Matrix is Zero
  • A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues
  • Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$
  • If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal
  • Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$
  • Are Linear Transformations of Derivatives and Integrations Linearly Independent?
  • Determine the Values of $a$ such that the 2 by 2 Matrix is Diagonalizable
  • A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
  • Diagonalize the 3 by 3 Matrix if it is Diagonalizable
  • All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$
  • Differentiating Linear Transformation is Nilpotent
  • Eigenvalues of Similarity Transformations
  • Inequality about Eigenvalue of a Real Symmetric Matrix
  • Null Space, Nullity, Range, Rank of a Projection Linear Transformation
  • If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix
  • Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix
  • The Rank of the Sum of Two Matrices
  • Dimension of the Sum of Two Subspaces
  • True or False. Every Diagonalizable Matrix is Invertible
  • True of False Problems on Determinants and Invertible Matrices
  • Subspace Spanned By Cosine and Sine Functions
  • Differentiation is a Linear Transformation
  • The Sum of Subspaces is a Subspace of a Vector Space
  • Idempotent Matrices are Diagonalizable
  • Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation
  • Union of Subspaces is a Subspace if and only if One is Included in Another
  • If $A$ is an Idempotent Matrix, then When $I-kA$ is an Idempotent Matrix?
  • Every Complex Matrix Can Be Written as $A=B+iC$, where $B, C$ are Hermitian Matrices
  • If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal
  • Determine All Matrices Satisfying Some Conditions on Eigenvalues and Eigenvectors
  • Find the Inverse Matrix Using the Cayley-Hamilton Theorem
  • Eigenvalues of Orthogonal Matrices Have Length 1. Every \times 3$ Orthogonal Matrix Has 1 as an Eigenvalue
  • There is at Least One Real Eigenvalue of an Odd Real Matrix
  • A Relation of Nonzero Row Vectors and Column Vectors
  • Express a Hermitian Matrix as a Sum of Real Symmetric Matrix and a Real Skew-Symmetric Matrix
  • Complex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues
  • Sequence Converges to the Largest Eigenvalue of a Matrix
  • Find All the Eigenvalues and Eigenvectors of the 6 by 6 Matrix
  • Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite
  • Positive definite Real Symmetric Matrix and its Eigenvalues
  • If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution
  • For Which Choices of $x$ is the Given Matrix Invertible?
  • If a Matrix is the Product of Two Matrices, is it Invertible?
  • Find a Linear Transformation Whose Image (Range) is a Given Subspace
  • Determine Whether Given Matrices are Similar
  • If Two Matrices are Similar, then their Determinants are the Same
  • Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)
  • Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$
  • Find the Nullity of the Matrix $A+I$ if Eigenvalues are
    , 2, 3, 4, 5$
  • Quiz 13 (Part 2) Find Eigenvalues and Eigenvectors of a Special Matrix
  • Quiz 13 (Part 1) Diagonalize a Matrix
  • Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix
  • Find the Formula for the Power of a Matrix
  • Common Eigenvector of Two Matrices $A, B$ is Eigenvector of $A+B$ and $AB$.
  • Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like.
  • Determinant of Matrix whose Diagonal Entries are 6 and 2 Elsewhere
  • Eigenvalues and Eigenvectors of Matrix Whose Diagonal Entries are 3 and 9 Elsewhere
  • Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$
  • Idempotent (Projective) Matrices are Diagonalizable
  • Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities
  • Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices
  • Determinant of a General Circulant Matrix
  • Compute Power of Matrix If Eigenvalues and Eigenvectors Are Given
  • Hyperplane Through Origin is Subspace of 4-Dimensional Vector Space
  • Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$
  • Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$
  • Determine a Value of Linear Transformation From $\R^3$ to $\R^2$
  • Basis of Span in Vector Space of Polynomials of Degree 2 or Less
  • Orthonormal Basis of Null Space and Row Space
  • Determine Whether Trigonometry Functions $\sin^2(x), \cos^2(x), 1$ are Linearly Independent or Dependent
  • True or False Problems of Vector Spaces and Linear Transformations
  • Quiz 11. Find Eigenvalues and Eigenvectors/ Properties of Determinants
  • Find All the Eigenvalues of Power of Matrix and Inverse Matrix
  • If Every Vector is Eigenvector, then Matrix is a Multiple of Identity Matrix
  • Quiz 10. Find Orthogonal Basis / Find Value of Linear Transformation
  • Prove the Cauchy-Schwarz Inequality
  • Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$
  • Hyperplane in $n$-Dimensional Space Through Origin is a Subspace
  • Coordinate Vectors and Dimension of Subspaces (Span)
  • Quiz 9. Find a Basis of the Subspace Spanned by Four Matrices
  • Condition that a Matrix is Similar to the Companion Matrix of its Characteristic Polynomial
  • Linearly Dependent if and only if a Vector Can be Written as a Linear Combination of Remaining Vectors
  • Give a Formula For a Linear Transformation From $\R^2$ to $\R^3$
  • 12 Examples of Subsets that Are Not Subspaces of Vector Spaces
  • If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix
  • Normal Nilpotent Matrix is Zero Matrix
  • Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation
  • Linear Transformation to 1-Dimensional Vector Space and Its Kernel
  • Quiz 8. Determine Subsets are Subspaces: Functions Taking Integer Values / Set of Skew-Symmetric Matrices
  • Idempotent Linear Transformation and Direct Sum of Image and Kernel
  • Determine linear transformation using matrix representation
  • Find the Formula for the Power of a Matrix Using Linear Recurrence Relation
  • Solve a Linear Recurrence Relation Using Vector Space Technique
  • Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
  • Problems and Solutions About Similar Matrices
  • If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set?
  • Is there an Odd Matrix Whose Square is $-I$?
  • Basis with Respect to Which the Matrix for Linear Transformation is Diagonal
  • Matrix of Linear Transformation with respect to a Basis Consisting of Eigenvectors
  • Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space
  • Find a Condition that a Vector be a Linear Combination
  • Intersection of Two Null Spaces is Contained in Null Space of Sum of Two Matrices
  • Solve Linear Recurrence Relation Using Linear Algebra (Eigenvalues and Eigenvectors)
  • Matrix Representation of a Linear Transformation of Subspace of Sequences Satisfying Recurrence Relation
  • Sequences Satisfying Linear Recurrence Relation Form a Subspace
  • Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$.
  • Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space
  • Given a Spanning Set of the Null Space of a Matrix, Find the Rank
  • If a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero Matrix
  • Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$
  • Find the Inverse Matrix of a \times 3$ Matrix if Exists
  • Express a Vector as a Linear Combination of Given Three Vectors
  • Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices
  • Solve the System of Linear Equations and Give the Vector Form for the General Solution
  • The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns
  • Every Plane Through the Origin in the Three Dimensional Space is a Subspace
  • The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero
  • Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation
  • Summary: Possibilities for the Solution Set of a System of Linear Equations
  • Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix
  • Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace
  • Linearly Independent vectors $\mathbf{v}_1, \mathbf{v}_2$ and Linearly Independent Vectors $A\mathbf{v}_1, A\mathbf{v}_2$ for a Nonsingular Matrix
  • Dual Vector Space and Dual Basis, Some Equality
  • Quiz 3. Condition that Vectors are Linearly Dependent/ Orthogonal Vectors are Linearly Independent
  • Find a Nonsingular Matrix Satisfying Some Relation
  • Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent
  • Determine Linearly Independent or Linearly Dependent. Express as a Linear Combination
  • Linear Transformation, Basis For the Range, Rank, and Nullity, Not Injective
  • The Inverse Matrix of an Upper Triangular Matrix with Variables
  • The Union of Two Subspaces is Not a Subspace in a Vector Space
  • Quiz 2. The Vector Form For the General Solution / Transpose Matrices. Math 2568 Spring 2017.
  • Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$
  • If a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero Matrix
  • Prove a Given Subset is a Subspace and Find a Basis and Dimension
  • Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even
  • Vector Form for the General Solution of a System of Linear Equations
  • Invertible Matrix Satisfying a Quadratic Polynomial
  • Idempotent Matrices. 2007 University of Tokyo Entrance Exam Problem
  • If matrix product $AB$ is a square, then is $BA$ a square matrix?
  • Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017.
  • Matrix $XY-YX$ Never Be the Identity Matrix
  • Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix
  • Determine a Matrix From Its Eigenvalue
  • Linear Combination of Eigenvectors is Not an Eigenvector
  • Use Cramer's Rule to Solve a \times 2$ System of Linear Equations
  • Basis and Dimension of the Subspace of All Polynomials of Degree 4 or Less Satisfying Some Conditions.
  • Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$
  • Find the Distance Between Two Vectors if the Lengths and the Dot Product are Given
  • True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces
  • Find a Matrix so that a Given Subset is the Null Space of the Matrix, hence it's a Subspace
  • The Inverse Matrix is Unique
  • Sherman-Woodbery Formula for the Inverse Matrix
  • Find Values of $a$ so that Augmented Matrix Represents a Consistent System
  • Condition that Two Matrices are Row Equivalent
  • Determine Null Spaces of Two Matrices
  • If Eigenvalues of a Matrix $A$ are Less than
    $, then Determinant of $I-A$ is Positive
  • Is a Set of All Nilpotent Matrix a Vector Space?
  • Orthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct Eigenvalues
  • Dimension of Null Spaces of Similar Matrices are the Same
  • Rotation Matrix in Space and its Determinant and Eigenvalues
  • Given Graphs of Characteristic Polynomial of Diagonalizable Matrices, Determine the Rank of Matrices
  • Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible.
  • Find the Inverse Matrix of a Matrix With Fractions
  • A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
  • How to Diagonalize a Matrix. Step by Step Explanation.
  • Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix
  • Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable
  • Eigenvalues of a Hermitian Matrix are Real Numbers
  • Cosine and Sine Functions are Linearly Independent
  • Maximize the Dimension of the Null Space of $A-aI$
  • Find Values of $h$ so that the Given Vectors are Linearly Independent
  • Compute Determinant of a Matrix Using Linearly Independent Vectors
  • Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$.
  • A Matrix Having One Positive Eigenvalue and One Negative Eigenvalue
  • Given All Eigenvalues and Eigenspaces, Compute a Matrix Product
  • Two Eigenvectors Corresponding to Distinct Eigenvalues are Linearly Independent
  • Is the Determinant of a Matrix Additive?
  • Eigenvalues of a Stochastic Matrix is Always Less than or Equal to 1
  • Eigenvalues of Squared Matrix and Upper Triangular Matrix
  • Eigenvalues of a Matrix and Its Squared Matrix
  • Linear Transformation and a Basis of the Vector Space $\R^3$
  • Given Eigenvectors and Eigenvalues, Compute a Matrix Product (Stanford University Exam)
  • Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix
  • Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam)
  • Idempotent Matrix and its Eigenvalues
  • Find All the Values of $x$ so that a Given \times 3$ Matrix is Singular
  • Find All Values of $x$ so that a Matrix is Singular
  • Subspace of Skew-Symmetric Matrices and Its Dimension
  • Vector Space of Polynomials and a Basis of Its Subspace
  • A Matrix Representation of a Linear Transformation and Related Subspaces
  • Inner Product, Norm, and Orthogonal Vectors
  • Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known
  • Linear Independent Continuous Functions
  • Vector Space of Polynomials and Coordinate Vectors
  • Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$
  • Linear Properties of Matrix Multiplication and the Null Space of a Matrix
  • Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$
  • Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis
  • Find a Basis for a Subspace of the Vector Space of \times 2$ Matrices
  • Any Vector is a Linear Combination of Basis Vectors Uniquely
  • A Basis for the Vector Space of Polynomials of Degree Two or Less and Coordinate Vectors
  • Nilpotent Matrices and Non-Singularity of Such Matrices
  • Subspaces of Symmetric, Skew-Symmetric Matrices
  • Find a Value of a Linear Transformation From $\R^2$ to $\R^3$
  • Linear Independent Vectors and the Vector Space Spanned By Them
  • Rank and Nullity of a Matrix, Nullity of Transpose
  • Sum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All Zero
  • How to Find the Determinant of the \times 3$ Matrix
  • Find a Basis and Determine the Dimension of a Subspace of All Polynomials of Degree $n$ or Less
  • Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
  • Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$
  • Subspaces of the Vector Space of All Real Valued Function on the Interval
  • Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?
  • Find a Basis For the Null Space of a Given \times 3$ Matrix
  • Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space
  • Find Values of $a$ so that the Matrix is Nonsingular
  • Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$
  • The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$
  • If Vectors are Linearly Dependent, then What Happens When We Add One More Vectors?
  • Subset of Vectors Perpendicular to Two Vectors is a Subspace
  • Express a Vector as a Linear Combination of Other Vectors
  • Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector
  • Symmetric Matrices and the Product of Two Matrices
  • A Condition that a Linear System has Nontrivial Solutions
  • 10 True or False Problems about Basic Matrix Operations
  • Find the Rank of a Matrix with a Parameter
  • Possibilities For the Number of Solutions for a Linear System
  • Determine When the Given Matrix Invertible
  • If the Matrix Product $AB=0$, then is $BA=0$ as Well?
  • True or False: $(A-B)(A+B)=A^2-B^2$ for Matrices $A$ and $B$
  • Quiz: Possibilities For the Solution Set of a Homogeneous System of Linear Equations
  • In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.
  • Find a Polynomial Satisfying the Given Conditions on Derivatives
  • Quiz: Linear Equations and Matrix Entreis
  • Companion Matrix for a Polynomial
  • If a Power of a Matrix is the Identity, then the Matrix is Diagonalizable
  • Isomorphism of the Endomorphism and the Tensor Product of a Vector Space
  • The Vector Space Consisting of All Traceless Diagonal Matrices
  • True or False Quiz About a System of Linear Equations
  • Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent?
  • Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix.
  • A Linear Transformation from Vector Space over Rational Numbers to itself
  • Exponential Functions are Linearly Independent
  • Conditions on Coefficients that a Matrix is Nonsingular
  • Matrix Representations for Linear Transformations of the Vector Space of Polynomials
  • Is an Eigenvector of a Matrix an Eigenvector of its Inverse?
  • Matrices Satisfying $HF-FH=-2F$
  • Matrices Satisfying the Relation $HE-EH=2E$
  • True or False: Eigenvalues of a Real Matrix Are Real Numbers
  • Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations
  • Solving a System of Linear Equations By Using an Inverse Matrix
  • A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues
  • If the Images of Vectors are Linearly Independent, then They Are Linearly Independent
  • Two Subspaces Intersecting Trivially, and the Direct Sum of Vector Spaces.
  • Projection to the subspace spanned by a vector
  • A Square Root Matrix of a Symmetric Matrix
  • Inequality Regarding Ranks of Matrices
  • Characteristic Polynomials of $AB$ and $BA$ are the Same
  • Perturbation of a Singular Matrix is Nonsingular
  • Simple Commutative Relation on Matrices
  • All the Eigenvectors of a Matrix Are Eigenvectors of Another Matrix
  • Find the Limit of a Matrix
  • Linearly Independent/Dependent Vectors Question
  • How to Calculate and Simplify a Matrix Polynomial
  • Trace of the Inverse Matrix of a Finite Order Matrix
  • Calculate Determinants of Matrices
  • Find a Matrix that Maps Given Vectors to Given Vectors
  • Find All Matrices Satisfying a Given Relation
  • Symmetric Matrix and Its Eigenvalues, Eigenspaces, and Eigenspaces
  • Calculate $A^{10}$ for a Given Matrix $A$
  • Find a Basis of the Subspace of All Vectors that are Perpendicular to the Columns of the Matrix
  • Given the Characteristic Polynomial of a Diagonalizable Matrix, Find the Size of the Matrix, Dimension of Eigenspace
  • If the Kernel of a Matrix $A$ is Trivial, then $A^T A$ is Invertible
  • Diagonalizable Matrix with Eigenvalue 1, -1
  • Find a Formula for a Linear Transformation
  • Find the Rank of the Matrix $A+I$ if Eigenvalues of $A$ are
    , 2, 3, 4, 5$
  • Stochastic Matrix (Markov Matrix) and its Eigenvalues and Eigenvectors
  • The Subspace of Matrices that are Diagonalized by a Fixed Matrix
  • Find all Values of x such that the Given Matrix is Invertible
  • Equivalent Conditions to be a Unitary Matrix
  • Finite Order Matrix and its Trace
  • Solve a System of Linear Equations by Gauss-Jordan Elimination
  • A Matrix is Invertible If and Only If It is Nonsingular
  • Properties of Nonsingular and Singular Matrices
  • Solving a System of Linear Equations Using Gaussian Elimination
  • How to Find Eigenvalues of a Specific Matrix.
  • If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent
  • Questions About the Trace of a Matrix
  • Linear Dependent/Independent Vectors of Polynomials
  • Possibilities of the Number of Solutions of a Homogeneous System of Linear Equations
  • Common Eigenvector of Two Matrices and Determinant of Commutator
  • Transpose of a Matrix and Eigenvalues and Related Questions
  • Nilpotent Matrix and Eigenvalues of the Matrix
  • Determinant/Trace and Eigenvalues of a Matrix
  • How to Find a Formula of the Power of a Matrix
  • Powers of a Diagonal Matrix
  • A Linear Transformation Maps the Zero Vector to the Zero Vector
  • Similar Matrices Have the Same Eigenvalues
  • Invertible Idempotent Matrix is the Identity Matrix
• Math-Magic
  • The Trick of a Mathematical Game. The One's Digit of the Sum of Two Numbers.
• Module Theory
  • A Module $M$ is Irreducible if and only if $M$ is isomorphic to $R/I$ for a Maximal Ideal $I$.
  • A Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as Generator
  • Finitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator
  • Nilpotent Ideal and Surjective Module Homomorphisms
  • Difference Between Ring Homomorphisms and Module Homomorphisms
  • Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure?
  • Submodule Consists of Elements Annihilated by Some Power of an Ideal
  • Ascending Chain of Submodules and Union of its Submodules
  • Linearly Dependent Module Elements / Module Homomorphism and Linearly Independency
  • Short Exact Sequence and Finitely Generated Modules
  • Annihilator of a Submodule is a 2-Sided Ideal of a Ring
  • Torsion Submodule, Integral Domain, and Zero Divisors
  • Basic Exercise Problems in Module Theory
• Probability
  • How to Prove Markov's Inequality and Chebyshev's Inequality
  • How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions
  • Expected Value and Variance of Exponential Random Variable
  • Condition that a Function Be a Probability Density Function
  • Conditional Probability When the Sum of Two Geometric Random Variables Are Known
  • Probability that Three Pieces Form a Triangle
  • Upper Bound of the Variance When a Random Variable is Bounded
  • Probability that Alice Tossed a Coin Three Times If Alice and Bob Tossed Totally 7 Times
  • Probability of Getting Two Red Balls From the Chosen Box
  • Coupon Collecting Problem: Find the Expectation of Boxes to Collect All Toys
  • Can a Student Pass By Randomly Answering Multiple Choice Questions?
  • Given the Variance of a Bernoulli Random Variable, Find Its Expectation
  • Expectation, Variance, and Standard Deviation of Bernoulli Random Variables
  • Probability that Alice Wins n Games Before Bob Wins m Games
  • Probabilities of An Infinite Sequence of Die Rolling
  • Interchangeability of Limits and Probability of Increasing or Decreasing Sequence of Events
  • Linearity of Expectations E(X+Y) = E(X) + E(Y)
  • Successful Probability of a Communication Network Diagram
  • Lower and Upper Bounds of the Probability of the Intersection of Two Events
  • Find the Conditional Probability About Math Exam Experiment
  • What is the Probability that Selected Coin was Two-Headed?
  • If a Smartphone is Defective, Which Factory Made It?
  • If At Least One of Two Coins Lands Heads, What is the Conditional Probability that the First Coin Lands Heads?
  • Probability of Having Lung Cancer For Smokers
  • Overall Fraction of Defective Smartphones of Three Factories
  • Complement of Independent Events are Independent
  • Independent and Dependent Events of Three Coins Tossing
  • Independent Events of Playing Cards
  • Jewelry Company Quality Test Failure Probability
  • What is the Probability that All Coins Land Heads When Four Coins are Tossed If...?
  • Pick Two Balls from a Box, What is the Probability Both are Red?
  • Conditional Probability Problems about Die Rolling
  • Probability Problems about Two Dice
• Ring theory
  • The Zero is the only Nilpotent Element of the Quotient Ring by its Nilradical
  • Three Equivalent Conditions for an Ideal is Prime in a PID
  • Every Prime Ideal of a Finite Commutative Ring is Maximal
  • Ring Homomorphisms and Radical Ideals
  • Example of an Element in the Product of Ideals that Cannot be Written as the Product of Two Elements
  • Is the Set of Nilpotent Element an Ideal?
  • Boolean Rings Do Not Have Nonzero Nilpotent Elements
  • If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian?
  • A Ring is Commutative if Whenever $ab=ca$, then $b=c$
  • If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field.
  • Is the Quotient Ring of an Integral Domain still an Integral Domain?
  • Polynomial Ring with Integer Coefficients and the Prime Ideal $I=\{f(x) \in \Z[x] \mid f(-2)=0\}$
  • A Ring Has Infinitely Many Nilpotent Elements if $ab=1$ and $ba \neq 1$
  • If $ab=1$ in a Ring, then $ba=1$ when $a$ or $b$ is Not a Zero Divisor
  • Every Ideal of the Direct Product of Rings is the Direct Product of Ideals
  • Every Prime Ideal in a PID is Maximal / A Quotient of a PID by a Prime Ideal is a PID
  • The Quotient Ring $\Z[i]/I$ is Finite for a Nonzero Ideal of the Ring of Gaussian Integers
  • The Image of an Ideal Under a Surjective Ring Homomorphism is an Ideal
  • No Nonzero Zero Divisor in a Field / Direct Product of Rings is Not a Field
  • Every Prime Ideal is Maximal if $a^n=a$ for any Element $a$ in the Commutative Ring
  • A ring is Local if and only if the set of Non-Units is an Ideal
  • The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$.
  • Is the Given Subset of The Ring of Integer Matrices an Ideal?
  • Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals
  • The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD)
  • The Quadratic Integer Ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD)
  • Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$
  • Idempotent Elements and Zero Divisors in a Ring and in an Integral Domain
  • The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain
  • Every Ring of Order $p^2$ is Commutative
  • The Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not Isomorphic
  • Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$
  • Every Integral Domain Artinian Ring is a Field
  • Three Equivalent Conditions for a Ring to be a Field
  • If $R$ is a Noetherian Ring and $f:R\to R'$ is a Surjective Homomorphism, then $R'$ is Noetherian
  • The Preimage of Prime ideals are Prime Ideals
  • The Inverse Image of an Ideal by a Ring Homomorphism is an Ideal
  • Polynomial $(x-1)(x-2)\cdots (x-n)-1$ is Irreducible Over the Ring of Integers $\Z$
  • If Two Ideals Are Comaximal in a Commutative Ring, then Their Powers Are Comaximal Ideals
  • Every Maximal Ideal of a Commutative Ring is a Prime Ideal
  • A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring
  • Irreducible Polynomial Over the Ring of Polynomials Over Integral Domain
  • Ring Homomorphisms from the Ring of Rational Numbers are Determined by the Values at Integers
  • Generators of the Augmentation Ideal in a Group Ring
  • There is Exactly One Ring Homomorphism From the Ring of Integers to Any Ring
  • Primary Ideals, Prime Ideals, and Radical Ideals
  • $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.
  • Polynomial $x^4-2x-1$ is Irreducible Over the Field of Rational Numbers $\Q$
  • Characteristic of an Integral Domain is 0 or a Prime Number
  • 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$
  • A Prime Ideal in the Ring $\Z[\sqrt{10}]$
  • If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain
  • How Many Solutions for $x+x=1$ in a Ring?
  • Ideal Quotient (Colon Ideal) is an Ideal
  • Non-Prime Ideal of Continuous Functions
  • The Ideal $(x)$ is Prime in the Polynomial Ring $R[x]$ if and only if the Ring $R$ is an Integral Domain
  • If the Quotient Ring is a Field, then the Ideal is Maximal
  • Finite Integral Domain is a Field
  • Ring of Gaussian Integers and Determine its Unit Elements
  • $\sqrt[m]{2}$ is an Irrational Number
  • The Ideal Generated by a Non-Unit Irreducible Element in a PID is Maximal
  • In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal
  • Equivalent Conditions For a Prime Ideal in a Commutative Ring
  • Prime Ideal is Irreducible in a Commutative Ring
  • Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal
  • Nilpotent Element a in a Ring and Unit Element
    -ab$
  • Rings \Z$ and \Z$ are Not Isomorphic
• Welcome
  • Welcome to Problems in Mathematics
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• Linear Algebra
  • Bases and Coordinate Vectors
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  • Dot Products and Length of Vectors
  • Eigenvalues and Eigenvectors of Linear Transformations
  • Eigenvectors and Eigenspaces
  • Elementary Row Operations
  • Gaussian-Jordan Elimination
  • General Vector Spaces
  • Introduction to Eigenvalues and Eigenvectors
  • Introduction to Matrices
  • Inverse Matrices
  • Jordan Canonical Form
  • Linear Combination and Linear Independence
  • Linear Transformation Between Vector Spaces
  • Linear Transformation from $\R^n$ to $\R^m$
  • Linearly Independency of General Vectors
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