$\sqrt[m]{2}$ is an Irrational Number
Problem 179
Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.
Add to solve later Author: Yu
Add to solve later Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam)
Problem 178
Let
\[\begin{bmatrix}
0 & 0 & 1 \\
1 &0 &0 \\
0 & 1 & 0
\end{bmatrix}.\]
(a) Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers?
(b) Calculate $A^{2009}$.
(Princeton University, Linear Algebra Exam)
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Add to solve later The Ideal Generated by a Non-Unit Irreducible Element in a PID is Maximal
Problem 177
Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element.
Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal.
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Add to solve later Idempotent Matrix and its Eigenvalues
Problem 176
Let $A$ be an $n \times n$ matrix. We say that $A$ is idempotent if $A^2=A$.
(a) Find a nonzero, nonidentity idempotent matrix.
(b) Show that eigenvalues of an idempotent matrix $A$ is either $0$ or $1$.
(The Ohio State University, Linear Algebra Final Exam Problem)
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Add to solve later In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal
Problem 175
Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$.
Show that $P$ is a maximal ideal in $R$.
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Add to solve later Equivalent Conditions For a Prime Ideal in a Commutative Ring
Problem 174
Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent:
(a) The ideal $P$ is a prime ideal.
(b) For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.
Add to solve later Prime Ideal is Irreducible in a Commutative Ring
Problem 173
Let $R$ be a commutative ring. An ideal $I$ of $R$ is said to be irreducible if it cannot be written as an intersection of two ideals of $R$ which are strictly larger than $I$.
Prove that if $\frakp$ is a prime ideal of the commutative ring $R$, then $\frakp$ is irreducible.
Add to solve later Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal
Problem 172
Let $R$ be a commutative ring.
Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$.
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Add to solve later Nilpotent Element a in a Ring and Unit Element $1-ab$
Problem 171
Let $R$ be a commutative ring with $1 \neq 0$.
An element $a\in R$ is called nilpotent if $a^n=0$ for some positive integer $n$.
Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$.
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Add to solve later Rings $2\Z$ and $3\Z$ are Not Isomorphic
Problem 170
Prove that the rings $2\Z$ and $3\Z$ are not isomorphic.
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Add to solve later Find All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular
Problem 169
Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
\[A=\begin{bmatrix}
x & x^2 & 1 \\
2 &3 &1 \\
0 & -1 & 1
\end{bmatrix}.\]
Add to solve later Find All Values of $x$ so that a Matrix is Singular
Problem 168
Let
\[A=\begin{bmatrix}
1 & -x & 0 & 0 \\
0 &1 & -x & 0 \\
0 & 0 & 1 & -x \\
0 & 1 & 0 & -1
\end{bmatrix}\]
be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.
Add to solve later Abelian Groups and Surjective Group Homomorphism
Problem 167
Let $G, G’$ be groups. Suppose that we have a surjective group homomorphism $f:G\to G’$.
Show that if $G$ is an abelian group, then so is $G’$.
Add to solve later Subspace of Skew-Symmetric Matrices and Its Dimension
Problem 166
Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.)
(a) Prove that the subset $W$ is a subspace of $V$.
(b) Find the dimension of $W$.
(The Ohio State University Linear Algebra Exam Problem)
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Add to solve later Vector Space of Polynomials and a Basis of Its Subspace
Problem 165
Let $P_2$ be the vector space of all polynomials of degree two or less.
Consider the subset in $P_2$
\[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\]
where
\begin{align*}
&p_1(x)=1, &p_2(x)=x^2+x+1, \\
&p_3(x)=2x^2, &p_4(x)=x^2-x+1.
\end{align*}
(a) Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.
(b) Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.
(c) For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.
(The Ohio State University Linear Algebra Exam Problem)
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Add to solve later A Matrix Representation of a Linear Transformation and Related Subspaces
Problem 164
Let $T:\R^4 \to \R^3$ be a linear transformation defined by
\[ T\left (\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \,\right) = \begin{bmatrix}
x_1+2x_2+3x_3-x_4 \\
3x_1+5x_2+8x_3-2x_4 \\
x_1+x_2+2x_3
\end{bmatrix}.\]
(a) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.
(b) Find a basis for the null space of $T$.
(c) Find the rank of the linear transformation $T$.
(The Ohio State University Linear Algebra Exam Problem)
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Add to solve later A Homomorphism from the Additive Group of Integers to Itself
Problem 163
Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism.
Then show that there exists an integer $a$ such that
\[f(n)=an\]
for any integer $n$.
Add to solve later Inner Product, Norm, and Orthogonal Vectors
Problem 162
Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ are vectors in $\R^n$. Suppose that vectors $\mathbf{u}_1$, $\mathbf{u}_2$ are orthogonal and the norm of $\mathbf{u}_2$ is $4$ and $\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number $a$ in $\mathbf{u_1}=\mathbf{u_2}+a\mathbf{u}_3$.
(The Ohio State University, Linear Algebra Exam Problem)
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Add to solve later Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup
Problem 161
Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$.
Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.
Add to solve later Finite Group and Subgroup Criteria
Problem 160
Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.
Then show that $H$ is a subgroup of $G$.
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