Problem 459

Let
$A=\begin{bmatrix} 1-a & a\\ -a& 1+a \end{bmatrix}$ be a $2\times 2$ matrix, where $a$ is a complex number.
Determine the values of $a$ such that the matrix $A$ is diagonalizable.

(Nagoya University, Linear Algebra Exam Problem)

Problem 458

Let $G$ be a finite group of order $217$.

(a) Prove that $G$ is a cyclic group.

(b) Determine the number of generators of the group $G$.

Problem 457

Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.

(a) Prove that for sufficiently small positive real $\epsilon$, the equation
$A\mathbf{x}+\epsilon\mathbf{x}=\mathbf{v}$ has a unique solution $\mathbf{x}=\mathbf{x}(\epsilon) \in \R^n$.

(b) Evaluate
$\lim_{\epsilon \to 0^+} \epsilon \mathbf{x}(\epsilon)$ in terms of $\mathbf{v}$, the eigenvectors of $A$, and the inner product $\langle\, ,\,\rangle$ on $\R^n$.

(University of California, Berkeley, Linear Algebra Qualifying Exam)

Problem 456

Determine whether the matrix
$A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}$ is diagonalizable.

If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

Problem 455

Let $G$ be a finite group.
The centralizer of an element $a$ of $G$ is defined to be
$C_G(a)=\{g\in G \mid ga=ag\}.$

A conjugacy class is a set of the form
$\Cl(a)=\{bab^{-1} \mid b\in G\}$ for some $a\in G$.

(a) Prove that the centralizer of an element of $a$ in $G$ is a subgroup of the group $G$.

(b) Prove that the order (the number of elements) of every conjugacy class in $G$ divides the order of the group $G$.

Problem 454

Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.

Problem 453

Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less.
Consider the differentiation linear transformation $T: P_n\to P_n$ defined by
$T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).$

(a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a basis of $P_2$. Find the matrix representation $A$ of the linear transformation $T$ with respect to the basis $B$.

(b) Compute $A^3$, where $A$ is the matrix obtained in part (a).

(c) If you computed $A^3$ in part (b) directly, then is there any theoretical explanation of your result?

(d) Now we consider the general case. Let $B$ be any basis of the vector space of $P_n$ and let $A$ be the matrix representation of the linear transformation $T$ with respect to the basis $B$.
Prove that without any calculation that the matrix $A$ is nilpotent.

Problem 452

Let $A$ be an $n\times n$ complex matrix.
Let $S$ be an invertible matrix.

(a) If $SAS^{-1}=\lambda A$ for some complex number $\lambda$, then prove that either $\lambda^n=1$ or $A$ is a singular matrix.

(b) If $n$ is odd and $SAS^{-1}=-A$, then prove that $0$ is an eigenvalue of $A$.

(c) Suppose that all the eigenvalues of $A$ are integers and $\det(A) > 0$. If $n$ is odd and $SAS^{-1}=A^{-1}$, then prove that $1$ is an eigenvalue of $A$.

Problem 451

Let $A$ be an $n\times n$ real symmetric matrix.
Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality
$\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.$

Problem 450

Let $\mathbf{u}=\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
$T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.$

(a) Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.

(b) Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.

(c) Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.

(d) Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.

(e) Let
$B=\left\{\, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \,\right\}$ be a basis for $\R^3$.
Calculate the coordinates of $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ with respect to $B$.

(The Ohio State University, Linear Algebra Exam Problem)

Problem 449

Let $R$ be a commutative ring with $1$ and let $M$ be an $R$-module.
Prove that the $R$-module $M$ is irreducible if and only if $M$ is isomorphic to $R/I$, where $I$ is a maximal ideal of $R$, as an $R$-module.

Problem 448

Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$.
The product of $H$ and $N$ is defined to be the subset
$H\cdot N=\{hn\in G\mid h \in H, n\in N\}.$ Prove that the product $H\cdot N$ is a subgroup of $G$.

Problem 447

Let $A$ be a square matrix such that
$A^{\trans}A=A,$ where $A^{\trans}$ is the transpose matrix of $A$.
Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.

Problem 446

Prove that there are infinitely many prime numbers in ONE-LINE.

Problem 445

Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism.
Suppose that $f:G\to H$ is bijective.
Then there exists a map $\psi:H\to G$ such that
$\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.$ Then prove that $\psi:H \to G$ is also a group homomorphism.

Problem 444

Let $G, G’$ be groups. Let $\phi:G\to G’$ be a group homomorphism.
Then prove that for any element $g\in G$, we have
$\phi(g^{-1})=\phi(g)^{-1}.$

Problem 443

Let $A=B=\Z$ be the additive group of integers.
Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$.

(a) Prove that $\phi$ is a group homomorphism.

(b) Prove that $\phi$ is injective.

(c) Prove that there does not exist a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi=\id_A$.

Problem 442

Consider the following system of linear equations
\begin{align*}
2x+3y+z&=-1\\
3x+3y+z&=1\\
2x+4y+z&=-2.
\end{align*}

(a) Find the coefficient matrix $A$ for this system.

(b) Find the inverse matrix of the coefficient matrix found in (a)

(c) Solve the system using the inverse matrix $A^{-1}$.

Problem 441

Let $A$ and $B$ be $m\times n$ matrices.
Prove that
$\rk(A+B) \leq \rk(A)+\rk(B).$

Problem 440

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.
Then prove that
$\dim(U+V) \leq \dim(U)+\dim(V).$