# Tagged: vector space

Vector Space Problems and Solutions.

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## Problem 541

Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$.
Consider a linear transformation $T:U\to V$.

Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).

## Problem 538

(a) Suppose that $A$ is an $n\times n$ real symmetric positive definite matrix.
Prove that
$\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}$ defines an inner product on the vector space $\R^n$.

(b) Let $A$ be an $n\times n$ real matrix. Suppose that
$\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}$ defines an inner product on the vector space $\R^n$.

Prove that $A$ is symmetric and positive definite.

## Problem 528

Let $V$ denote the vector space of all real $2\times 2$ matrices.
Suppose that the linear transformation from $V$ to $V$ is given as below.
$T(A)=\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}A-A\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}.$ Prove or disprove that the linear transformation $T:V\to V$ is an isomorphism.

## Problem 481

Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less.
Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where
\begin{align*}
\end{align*}

(a) Find a basis of $P_2$ among the vectors of $S$. (Explain why it is a basis of $P_2$.)

(b) Let $B’$ be the basis you obtained in part (a).
For each vector of $S$ which is not in $B’$, find the coordinate vector of it with respect to the basis $B’$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 480

(a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying
$2x+4y+3z+7w+1=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(b) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying
$2x+4y+3z+7w=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(These two problems look similar but note that the equations are different.)

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 463

Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation).
Let $V$ be the vector space of all linear transformations from $W$ to $W$.
The addition and the scalar multiplication of $V$ are given by those of linear transformations.

Let $T_1, T_2, T_3$ be the elements in $V$ defined by
\begin{align*}
T_1\left(\, f(x) \,\right)&=\frac{\mathrm{d}}{\mathrm{d}x}f(x)\6pt] T_2\left(\, f(x) \,\right)&=\frac{\mathrm{d}^2}{\mathrm{d}x^2}f(x)\\[6pt] T_3\left(\, f(x) \,\right)&=\int_{0}^x \! f(t)\,\mathrm{d}t. \end{align*} Then determine whether the set \{T_1, T_2, T_3\} are linearly independent or linearly dependent. ## All Linear Transformations that Take the Line y=x to the Line y=-x ## 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. ## Differentiating Linear Transformation is Nilpotent ## 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 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
$\dim(U+V) \leq \dim(U)+\dim(V).$

## Problem 435

Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
$\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.$ We put
$V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.$

(a) Prove that the map $f$ is a linear transformation.

(b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.

(c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
(This yields an isomorphism of $\R^2$ and $V$.)

(d) Define a map $g:V \to V$ by
$g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.$ Prove that the map $g$ is a linear transformation.

(e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.

(Kyoto University, Linear Algebra exam problem)

## Problem 433

Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients.

(a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by
$T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)$ for any $f(x)\in P_3$ is a linear transformation.

(b) Let $B=\{1, x, x^2, x^3\}$ be a basis of $P_3$. With respect to the basis $B$, find the matrix representation of the linear transformation $T$ in part (a).

## Problem 430

Let $V$ be a vector space over a field $K$.
If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset
$W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}$ is a subspace of the vector space $V$.

## Problem 429

Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.

## Problem 428

Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix
$A=\begin{bmatrix} 1 & 0 & 2 \\ 0 &3 &0 \\ 4 & 0 & 5 \end{bmatrix}.$

(a) Prove that the linear transformation $T$ sends points on the $x$-$z$ plane to points on the $x$-$z$ plane.

(b) Prove that the restriction of $T$ on the $x$-$z$ plane is a linear transformation.

(c) Find the matrix representation of the linear transformation obtained in part (b) with respect to the standard basis
$\left\{\, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \,\right\}$ of the $x$-$z$ plane.

## Problem 427

Let $W_1, W_2$ be subspaces of a vector space $V$. Then prove that $W_1 \cup W_2$ is a subspace of $V$ if and only if $W_1 \subset W_2$ or $W_2 \subset W_1$.

## Problem 400

Find all the eigenvalues and eigenvectors of the matrix
$A=\begin{bmatrix} 10001 & 3 & 5 & 7 &9 & 11 \\ 1 & 10003 & 5 & 7 & 9 & 11 \\ 1 & 3 & 10005 & 7 & 9 & 11 \\ 1 & 3 & 5 & 10007 & 9 & 11 \\ 1 &3 & 5 & 7 & 10009 & 11 \\ 1 &3 & 5 & 7 & 9 & 10011 \end{bmatrix}.$

(MIT, Linear Algebra Homework Problem)

## Problem 395

Suppose that the vectors
$\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0 \\ -3 \\ -2 \\ 1 \end{bmatrix}$ are a basis vectors for the null space of a $4\times 5$ matrix $A$. Find a vector $\mathbf{x}$ such that
$\mathbf{x}\neq0, \quad \mathbf{x}\neq \mathbf{v}_1, \quad \mathbf{x}\neq \mathbf{v}_2,$ and
$A\mathbf{x}=\mathbf{0}.$

(Stanford University, Linear Algebra exam problem)

## Problem 392

Let $V$ be the subspace of $\R^4$ defined by the equation
$x_1-x_2+2x_3+6x_4=0.$ Find a linear transformation $T$ from $\R^3$ to $\R^4$ such that the null space $\calN(T)=\{\mathbf{0}\}$ and the range $\calR(T)=V$. Describe $T$ by its matrix $A$.

## Problem 377

Let $A$ be an $n\times n$ idempotent complex matrix.
Then prove that $A$ is diagonalizable.