## 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 $y=-x$.

Add to solve laterof the day

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

Add to solve later 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.

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

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

\[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.\]

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*)

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.

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 $A^{-1}$.

Let $A$ and $B$ be $m\times n$ matrices.

Prove that

\[\rk(A+B) \leq \rk(A)+\rk(B).\]

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.

Then prove that

\[\dim(U+V) \leq \dim(U)+\dim(V).\]

Determine whether each of the following statements is True or False.

**(a)** If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.

**(b)** If the characteristic polynomial of an $n \times n$ matrix $A$ is

\[p(\lambda)=(\lambda-1)^n+2,\]
then $A$ is invertible.

**(c)** If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.

**(d)** If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

**(e)** If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.

(Stanford University, Linear Algebra Exam Problem)

Read solution

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)

Add to solve laterLet $P_3$ 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).

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 $V$.

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

Add to solve later 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.

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 $W_2 \subset W_1$.

Add to solve laterA square matrix $A$ is called **idempotent** if $A^2=A$.

**(a)** Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix.

**(b)** Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then determine all integers $k$ such that the matrix $I-kA$ is idempotent.

**(c)** Let $A$ and $B$ be $n\times n$ matrices satisfying

\[AB=A \text{ and } BA=B.\]
Then prove that $A$ is an idempotent matrix.

**(a)** Prove that each complex $n\times n$ matrix $A$ can be written as

\[A=B+iC,\]
where $B$ and $C$ are Hermitian matrices.

**(b)** Write the complex matrix

\[A=\begin{bmatrix}

i & 6\\

2-i& 1+i

\end{bmatrix}\]
as a sum $A=B+iC$, where $B$ and $C$ are Hermitian matrices.

Let $A$ and $B$ be $n\times n$ matrices.

Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$.

Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$.