# Introduction to Linear Algebra

Some problems and solutions by the topics that are taught in the undergraduate linear algebra course (Math 2568) in the Ohio State University.

The number of chapters/sections are based on the textbook Introduction to Linear Algebra, 5th edition,

by L.W. Johnson, R.D. Riess, and J.T. Arnold.

Contents

- Chapter 1. Matrices and Systems of Linear Equations
- 1.1 Introduction to Matrices and Systems of linear equations
- 1.2 Echelon Form and Gaussian-Jordan Elimination
- 1.3 Consistent Systems of linear Equations
- 1.5 Matrix Operations
- 1.6 Algebraic Properties of Matrix operations
- 1.7 Linear Independence and Nonsingular Matrices
- 1.9 Matrix Inverses and Their Properties

- Midterm Exam 1 (covers Chapter 1)
- Chapter 2. Vectors in 2-Space and 3-Space
- Chapter 3. The Vector Space $\R^n$
- Chapter 4. The Eigenvalue Problem
- 4.1 The Eigenvalue Problem for 2×2 Matrices
- 4.2 Determinants and the Eigenvalue Problem
- 4.3 Elementary Operations and Determinants
- 4.4 Eigenvalues and Characteristic Polynomial
- 4.5 Eigenvectors and Eigenspaces
- 4.6 Complex Eigenvalues and Eigenvectors
- 4.7 Similarity Transformations and Diagonalization
- Extra Problems on Eigenvalues and Diagonalization

- Chapter 5. Vector Spaces and Linear Transformations
- Chapter 6. Determinants

The problems with ★ are more advanced problems than typical exercise problems, but not necessarily unaccessible.

## Chapter 1. Matrices and Systems of Linear Equations

### 1.1 Introduction to Matrices and Systems of linear equations

### 1.2 Echelon Form and Gaussian-Jordan Elimination

- Solving a system of linear equations using Gaussian elimination
- Find a polynomial satisfying the given conditions
- Solve a system of linear equations by Gauss-Jordan elimination

### 1.3 Consistent Systems of linear Equations

- Summary: possibilities for the solution set of a system of linear equations
- Find the rank of a matrix with a parameter
- Quiz: possibilities for the solution set of a homogeneous system of linear equations
- True or False quiz about a system of linear equations
- Quiz:Possibilities of the number of solutions of a homogeneous system of linear equations
- Find values of $a$ so that augmented matrix represents a consistent system
- The possibilities for the number of solutions of systems of linear equations that have more equations than unknowns

### 1.5 Matrix Operations

- If matrix product $AB$ is a square, then is $BA$ a square matrix?
- Symmetric matrices and the product of two matrices
- 10 True or False Problems about Basic Matrix Operations
- 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 $B$
- Matrix $XY-YX$ never be the identity matrix
- Vector form for the general solution of a system of linear equations
- Solve the system of linear equations and give the vector form for the general solution
- Example of a nilpotent matrix $A$ such that $A^2\neq O$ but $A^3=O$.
- Find the formula for the power of a matrix
- If $A$ is an idempotent matrix, then when $I-kA$ is an idempotent matrix?
- A Matrix Commuting With a Diagonal Matrix with Distinct Entries is Diagonal
- The Powers of the Matrix with Cosine and Sine Functions
- An Example of Matrices $A$, $B$ such that $\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B)$
- ★Is the product of a nilpotent matrix and an invertible matrix nilpotent?

### 1.6 Algebraic Properties of Matrix operations

- Compute the product $A^{2017}\mathbf{u}$ of a matrix power and a vector
- Find the distance between two vectors if the lengths and the dot product are given
- Find all matrices $B$ that commutes with a given matrix $A$: $AB=BA$
- Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$
- 7 Problems on Skew-Symmetric Matrices

### 1.7 Linear Independence and Nonsingular Matrices

- Find values of $a$ so that the matrix is nonsingular
- If vectors are linearly dependent, then what happens when we add one more vectors?
- Express a vector as a linear combination of other vectors
- Possibilities for the number of solutions for a linear system
- Linearly independent/dependent vectors question
- Properties of nonsingular and singular matrices
- Find values of $h$ so that the given vectors are linearly independent
- If a matrix $A$ is singular, there exists nonzero $B$ such that the product $AB$ is the zero matrix
- Determine linearly independent or linearly dependent. Express as a linear combination
- Determine conditions on scalars so that the set of vectors is linearly dependent
- Determine a Condition on $a$ so that Vectors are Linearly Dependent
- Linearly (in)dependent vectors $\mathbf{v}_1, \mathbf{v}_2$ and linearly (in)dependent vectors $A\mathbf{v}_1, A\mathbf{v}_2$ for a (nonsingular) matrix
- If a matrix $A$ is singular, then exists nonzero $B$ such that $AB$ is the zero matrix
- The Transpose of a Nonsingular Matrix is Nonsingular
- The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$.

### 1.9 Matrix Inverses and Their Properties

- Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
- Invertible idempotent matrix is the identity matrix
- Determine when the given matrix invertible.
- Determine whether the following matrix invertible. If so find its inverse matrix.
- 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 matrix is invertible if and only if it is nonsingular
- Powers of a diagonal matrix
- Sherman-Woodbery formula for the inverse matrix
- The inverse matrix is unique
- Idempotent matrices. 2007 University of Tokyo entrance exam problem
- Invertible matrix satisfying a quadratic polynomial
- The inverse matrix of an upper triangular matrix with variables
- Find a nonsingular matrix satisfying some relation
- Solve a system by the inverse matrix and compute $A^{2017}\mathbf{x}$
- Find the inverse matrix of a $3\times 3$ matrix if exists
- If a matrix is the product of two matrices, is it invertible?
- Solve the system of linear equations using the inverse matrix of the coefficient matrix
- The Inverse Matrix of the Transpose is the Transpose of the Inverse Matrix
- Construction of a Symmetric Matrix whose Inverse Matrix is Itself
- A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors
- Two Matrices are Nonsingular if and only if the Product is Nonsingular
- Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible?
- The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive

## Midterm Exam 1 (covers Chapter 1)

- Linear Algebra Midterm 1 at the Ohio State University (1/3)
- Linear Algebra Midterm 1 at the Ohio State University (2/3)
- Linear Algebra Midterm 1 at the Ohio State University (3/3)

## Chapter 2. Vectors in 2-Space and 3-Space

### 2.1 Vectors in The Plane

### 2.2 Vectors in Space

### 2.3 The Dot Product and The Cross Product

## Chapter 3. The Vector Space $\R^n$

### 3.2 Vector Space Properties of $\R^n$

- The union of two subspaces is not a subspace in a vector space
- Union of subspaces is a subspace if and only if one is included in another
- The Intersection of Two Subspaces is also a Subspace

### 3.3 Examples of Subspaces

- The null space (the kernel) of a matrix is a subspace of ℝ$\R^n$
- Determine null spaces of two matrices
- Non-example of a subspace in 3-dimensional vector space $\R^3$
- Subset of vectors perpendicular to two vectors is a subspace
- Linear properties of matrix multiplication and the null space of a matrix
- Find a matrix so that a given subset is the null space of the matrix, hence it’s a subspace
- True or False. The intersection of bases is a basis of the intersection of subspaces
- The subset consisting of the zero vector is a subspace and its dimension is zero
- Every plane in the three dimensional space is a subspace
- Intersection of two null spaces is contained in null space of sum of two matrices
- Find a condition that a vector be a linear combination
- 10 examples of subsets that are not subspaces of vector spaces
- Hyperplane in n-dimensional space through origin is a subspace
- Hyperplane through origin is subspace of 4-dimensional vector space

### 3.4 Bases for Subspaces

- Find a basis and the dimension of the subspace of the 4-dimensional vector space
- Row equivalent matrix, bases for the null space, range, and row space of a matrix
- If two vectors satisfy $A\mathbf{x}=0$ then find another solution
- Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.
- Determine Whether Each Set is a Basis for $\R^3$
- ★Find a basis of the subspace of all vectors that are perpendicular to the columns of the matrix

### 3.5 Dimension

- Find a basis for the null space of a given 2×3 matrix
- Rank of the product of matrices $AB$ is less than or equal to the rank of A
- Two subspaces intersecting trivially, and the direct sum of vector spaces.
- Given a spanning set of the null space of a matrix, find the rank
- Dimension of the sum of two subspaces
- The rank of the sum of two matrices
- If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent
- Every Basis of a Subspace Has the Same Number of Vectors
- Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors
- The Column Vectors of Every $3\times 5$ Matrix Are Linearly Dependent
- Vector Space of 2 by 2 Traceless Matrices
- ★Rank and nullity of a matrix, nullity of transpose
- ★Column rank = row rank. (The rank of a matrix is the same as the rank of its transpose)

### 3.6 Orthogonal Bases for Subspaces

- Inner product, norm, and orthogonal vectors
- Prove the Cauchy-Schwarz inequality
- Orthonormal basis of null space and row space
- Orthogonal Nonzero Vectors Are Linearly Independent
- Find an Orthonormal Basis of $\R^3$ Containing a Given Vector
- Find an Orthonormal Basis of the Given Two Dimensional Vector Space

### 3.7 Linear Transformation from $\R^n$ to $\R^m$

- Find a value of a linear transformation from $\R^2$ to $\R^3$
- Give the formula for a linear transformation from $\R^3$ to $\R^2$
- Find a formula for a linear transformation
- Give a formula for a linear transformation if the values on basis vectors are known
- If the images of vectors are linearly independent, then they are linearly independent
- Projection to the subspace spanned by a vector
- Find a matrix that maps given vectors to given vectors
- A linear transformation maps the zero vector to the zero vector
- Range, null space, rank, and nullity of a linear transformation from $\R^2$ to $\R^3$
- A matrix representation of a linear transformation and related subspaces
- Linear transformation and a basis of the vector space $\R^3$
- Matrix representation of a linear transformation of the vector space $R^2$ to $R^2$
- Determine linear transformation using matrix representation
- Linear transformation to 1-dimensional vector space and its kernel
- Give a formula for a linear transformation from $\R^2$ to $\R^3$
- Find a general formula of a linear transformation from $\R^2$ to $\R^3$
- 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 value of linear transformation from $\R^3$ to $\R^2$
- Find a linear transformation whose image (range) is a given subspace
- Null Space, Nullity, Range, Rank of a Projection Linear Transformation
- All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$
- The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane
- An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism
- Linear Transformation that Maps Each Vector to Its Reflection with Respect to $x$-Axis
- Linear Transformation $T:\R^2 \to \R^2$ Given in Figure

## Chapter 4. The Eigenvalue Problem

### 4.1 The Eigenvalue Problem for 2×2 Matrices

- Determine a matrix from its eigenvalue
- An Example of a Real Matrix that Does Not Have Real Eigenvalues

### 4.2 Determinants and the Eigenvalue Problem

- Calculate determinants of matrices
- Find all values of $x$ such that the given matrix is invertible
- Nilpotent matrices and non-singularity of such matrices
- Find all values of $x$ so that a matrix is singular.
- Find all the values of $x$ so that a given 3×3 matrix is singular
- Compute determinant of a matrix using linearly independent vectors
- Find values of $h$ so that the given vectors are linearly independent
- Rotation matrix in space and its determinant and eigenvalues
- For which choices of $x$ is the given matrix invertible?
- True of False problems on determinants and invertible matrices

### 4.3 Elementary Operations and Determinants

(This section is not covered in Math 2568.)

- How to find the determinant of the 3×3 matrix
- Companion matrix for a polynomial
- The Determinant of a Skew-Symmetric Matrix is Zero

### 4.4 Eigenvalues and Characteristic Polynomial

- How to calculate and simplify a matrix polynomial
- Is an eigenvector of a matrix an eigenvector of its inverse?
- Find the nullity of the matrix $A+I$ if eigenvalues are $1, 2, 3, 4, 5$
- Find the rank of the matrix $A+I$ if eigenvalues of $A$ are 1,2,3,4,5
- Transpose of a matrix and eigenvalues and related questions.
- Similar matrices have the same eigenvalues
- Given eigenvectors and eigenvalues, compute a matrix product
- Eigenvalues of a matrix and its squared matrix
- Eigenvalues of a stochastic matrix is always less than or equal to 1
- A matrix having one positive eigenvalue and one negative eigenvalue
- Eigenvalues and their algebraic multiplicities of a matrix with a variable
- Linear combination of eigenvectors is not an eigenvector
- Find all the eigenvalues of power of matrix and inverse matrix
- Common eigenvector of two matrices $A, B$ is eigenvector of $A+B$ and $AB$.
- Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$
- Find the Inverse Matrix Using the Cayley-Hamilton Theorem
- How to use the Cayley-Hamilton Theorem to Find the Inverse Matrix
- The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$
- Eigenvalues of a Matrix and its Transpose are the Same
- Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials
- ★Matrices satisfying $HF-FH=-2F$
- ★Characteristic polynomials of AB A B and BA B A are the same.
- ★All the eigenvectors of a matrix are eigenvectors of another matrix

### 4.5 Eigenvectors and Eigenspaces

- Compute power of matrix if eigenvalues and eigenvectors are given
- Eigenvalues and algebraic/geometric multiplicities of matrix $A+cI$
- Eigenvalues and eigenvectors of matrix whose diagonal entries are 3 and 9 elsewhere
- Unit Vectors and Idempotent Matrices

### 4.6 Complex Eigenvalues and Eigenvectors

- True or False: Eigenvalues of a real matrix are real numbers.
- Matrices satisfying the relation HE-EH=2E
- Eigenvalues of a Hermitian matrix are real numbers
- Express a Hermitian matrix as a sum of real symmetric matrix and a real skew-symmetric matrix
- Every complex matrix can be written as $A=B+iC$, where $B, C$ are Hermitian matrices
- There is at least one real eigenvalue of an odd real matrix
- Eigenvalues of orthogonal matrices have length 1. Every $3\times 3$ orthogonal matrix has 1 as an eigenvalue
- Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors

### 4.7 Similarity Transformations and Diagonalization

- If two matrices are similar, then their determinants are the same
- Determine whether given matrices are similar
- Problems and solutions about similar matrices
- How to diagonalize a matrix. Step by step explanation
- Two matrices with the same characteristic polynomial. Diagonalize if possible.
- Eigenvalues of squared matrix and upper triangular matrix
- A square root matrix of a symmetric matrix with non-negative eigenvalues
- Diagonalizable matrix with eigenvalue 1, -1
- How to find a formula of the power of a matrix
- Diagonalizable by an orthogonal matrix implies a symmetric matrix
- A matrix similar to a diagonalizable matrix is also diagonalizable
- True or False. Every diagonalizable matrix is invertible
- Determine the Values of $a$ such that the 2 by 2 Matrix is Diagonalizable
- A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix

### Extra Problems on Eigenvalues and Diagonalization

- ★Square root of a diagonal matrix. How many square roots exist?
- ★Find all matrices satisfying a given relation
- Symmetric matrix and its eigenvalues, eigenspaces, and eigenspaces
- Given the characteristic polynomial of a diagonalizable matrix, find the size of the matrix, dimension of eigenspace
- Stochastic matrix (Markov matrix) and its eigenvalues and eigenvectors
- The subspace of matrices that are diagonalized by a fixed matrix.
- How to find eigenvalues of a specific matrix.
- Common eigenvector of two matrices and determinant of commutator
- Characteristic polynomial, eigenvalues, diagonalization problem
- Given graphs of characteristic polynomial of diagonalizable matrices, determine the rank of matrices
- Find the formula for the power of a matrix using linear recurrence relation
- Normal nilpotent matrix is zero matrix
- If 2 by 2 matrices satisfy $A=AB-BA$, then $A^2$ is zero matrix
- Condition that a matrix is similar to the companion matrix of its characteristic polynomial
- Determinant of a general circulant matrix
- Prove that the length $\|A^n\mathbf{v}\|$ is as small as we like.
- Determine dimensions of eigenspaces from characteristic polynomial of diagonalizable matrix
- Trace, determinant, and eigenvalue (Harvard University exam problem)
- Positive definite real symmetric matrix and its eigenvalues
- Inverse matrix of positive-definite symmetric matrix is positive-definite
- Inequality about Eigenvalue of a Real Symmetric Matrix
- Eigenvalues of Similarity Transformations
- If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal
- A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues
- Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$
- True or False: If $A, B$ are 2 by 2 Matrices such that $(AB)^2=O$, then $(BA)^2=O$
- Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix
- Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix
- Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues
- ★An Example of a Matrix that Cannot Be a Commutator
- ★If Matrices Commute $AB=BA$, then They Share a Common Eigenvector

## Chapter 5. Vector Spaces and Linear Transformations

### 5.2 Vector Spaces

### 5.3 Subspaces

- Find a basis and determine the dimension of a subspace of all polynomials of degree n or less
- Subspaces of the vector space of all real valued function on the interval
- Subspaces of symmetric, skew-symmetric matrices
- The Subspace of Linear Combinations whose Sums of Coefficients are zero
- Determine whether a set of functions $f(x)$ such that $f(x)=f(1-x)$ is a subspace
- Sequences satisfying linear recurrence relation form a subspace
- Is the Set of All Orthogonal Matrices a Vector Space?

### 5.4 Linear Independence, Bases, and Coordinates

- Linear independent continuous functions
- Linear independent vectors and the vector space spanned by them
- cosine and sine functions are linearly independent
- Exponential functions are linearly independent
- The vector space consisting of all traceless diagonal matrices
- Linear dependent/independent vectors of polynomials
- A basis for the vector space of polynomials of degree two or less and coordinate vectors
- Any vector is a linear combination of basis vectors uniquely
- Find a basis for a subspace of the vector space of $2\times 2$ matrices
- Show the subset of the vector space of polynomials is a subspace and find its basis
- Vector space of polynomials and a basis of its subspace
- Vector space of polynomials and coordinate vectors
- The set of $2\times 2$ Symmetric Matrices is a Subspace
- Subspace of skew-symmetric matrices and its dimension
- Basis and dimension of the subspace of all polynomials of degree 4 or less satisfying some conditions.
- Basis for subspace consisting of matrices commute with a given diagonal matrix
- Linearly dependent if and only if a vector can be written as a linear combination of remaining vectors
- Coordinate vectors and dimension of subspaces (span)
- Determine whether trigonometry functions $\sin^2(x), \cos^2(x), 1$ are linearly independent or dependent
- Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less
- Exponential Functions Form a Basis of a Vector Space
- Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$
- ★A linear transformation from vector space over rational numbers to itself

The following sections are not covered in Math 2568 at OSU.

### 5.6 Inner-Product Spaces, Orthogonal Bases, and Projections

- A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space
- The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization
- The Sum of Cosine Squared in an Inner Product Space

### 5.7 Linear Transformations

- Linear transformation, basis for the range, rank, and nullity, not injective
- Matrix representations for linear transformations of the vector space of polynomials.
- Matrix representation of a linear transformation of subspace of sequences satisfying recurrence relation
- Idempotent linear transformation and direct sum of image and kernel
- Linear transformation $T(X)=AX-XA$ and determinant of matrix representation
- Restriction of a linear transformation on the x-z plane is a linear transformation
- Subspace spanned by cosine and sine functions
- Differentiating Linear Transformation is Nilpotent
- A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero
- A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$
- Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism
- Eigenvalues and Eigenvectors of The Cross Product Linear Transformation

### 5.8 Operations with Linear Transformations

- Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism?
- Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$

### 5.10 Change of Basis and Diagonalization.

- Matrix of linear transformation with respect to a basis consisting of eigenvectors
- Basis with respect to which the matrix for linear transformation is diagonal
- Solve linear recurrence relation using linear algebra (eigenvalues and eigenvectors)2017/02/19

## Chapter 6. Determinants

### 6.4 Cramer’s Rule

### 6.5 Applications of Determinants: Inverses and Wronksians

- Find Inverse Matrices Using Adjoint Matrices
- Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$
- Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent