A transformation T: $V_n + V_n$ is described as, T rotates every point through the same angle $\phi$ about the origin. That is, T maps a point with polar coordinates (r, 0) onto the point with polar coordinates $(r, \theta+\phi)$, where $\phi$ is fixed. Also, Tmaps 0 onto itself. Determine whether T is linear. If T is linear, describe its null space and range, and compute its nullity and rank.

(a) Prove that a $2 \times 2$ matrix A commutes with every $2 \times 2$ matrix if and only if A commutes with each of the four matrices

$$\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right], \quad\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right], \quad\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right], \quad\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right] .$$

(b) Find all such matrices A.

A function $T: V_2 \rightarrow V_2$ is defined by the formula T(x, y)=(x+1, y+1), where (x,y) is an arbitrary point in $V_2$. Determine whether T is one-to-one on $V_2$. If it is, describe its range $T\left(V_2\right)$; for each point (u, v) in $T\left(V_2\right)$, let $(x, y)=T^{-1}(u, v)$ and give formulas for determining x and y in terms of u and v.

A transformation T: $V_n + V_n$ is defined by the formula T(x, y)=(2 x-y, x+y), where (x, y) is an arbitrary point in $V_n$ . Determine whether T is linear. If T is linear, describe its null space and range, and compute its nullity and rank.

Find all $2 \times 2$ matrices A such that $A^2=0$.

(a) Determine all solutions of the system

5x+2y−6z+2u=-1

x-y+z-u =-2 .

(b) Determine all solutions of the system

5 x+2 y-6 z+2 u=-1

x-y+z-$\mathrm{u}$=-2

x+y+z=6 .

Hadamard matrices, named for Jacques Hadamard (1865-1963), are those $n \times n$ matrices with the following properties:

I. Each entry is 1 or $-1$.

II. Each row, considered as a vector in $V_n$, has length $\sqrt{n}$.

III. The dot product of any two distinct rows is 0 .

Hadamard matrices arise in certain problems in geometry and the theory of numbers, and they have been applied recently to the construction of optimum code words in space communication. In spite of their apparent simplicity, they present many unsolved problems. The main unsolved problem at this time is to determine all $n$ for which an $n \times n$ Hadamard matrix exists. This exercise outlines a partial solution.

(a) Determine all $2 \times 2$ Hadamard matrices (there are exactly 8).

(b) This part of the exercise outlines a simple proof of the following theorem: If $A$ is an $n x n$ Hadamard matrix, where $n>2$, then $n$ is a multiple of 4 . The proof is based on two very simple lemmas concerning vectors in $n$-space. Prove each of these lemmas and apply them to the rows of Hadamard matrix to prove the theorem.

Let V and W be linear spaces, each with dimension 2 and each with basis $\left(e_1, e_2\right)$. Let $T: V \rightarrow W$ be a linear transformation such that

$$T\left(e_1+e_2\right)=3 e_1+9 e_2, \quad T\left(3 e_1+2 e_2\right)=7 e_1+23 e_2 .$$

(a) Compute $\boldsymbol{T}\left(e, e_1\right)$ and determine the nullity and rank of T.

(b) Determine the matrix of T relative to the given basis.

(c) Use the basis $\left(e_1, \mathrm{e},\right)$ for V and find a new basis of the form $\left(e_1+a e, 2 e_1+b e_2\right)$ for W, relative to which the matrix of T will be in diagonal form.

If

$$A=\left[\begin{array}{rrr}1 & -4 & 2 \\ -1 & 4 & -2\end{array}\right], B=\left[\begin{array}{rr}1 & 2 \\ -1 & 3 \\ 5 & -2\end{array}\right], C=\left[\begin{array}{rr}2 & 2 \\ 1 & -1 \\ 1 & -3\end{array}\right]$$

, compute B+C, AB, BA, AC, CA, A(2 B-3 C)

If a square matrix has a row of zeros or a column of zeros, prove that it is singular .

Let V={0,1}. Describe all functions $T: V \rightarrow V$. There are four altogether. Label them as $T_1, T_2, T_3, T_4$ and make a multiplication table showing the composition of each pair. Indicate which functions are one-to-one on V and give their inverses .

Determine the matrix of each of the following linear transformations of $V_n$ into $V_n$ :

(a) the identity transformation,

(b) the zero transformation,

(c) multiplication by a fixed scalar c.

A transformation T: $V_n + V_n$ is defined by the formula T(x, y)=(y, x), where (x, y) is an arbitrary point in $V_n$ . Determine whether T is linear. If T is linear, describe its null space and range, and compute its nullity and rank.

Apply the Gauss-Jordan elimination process to the following system. If a solution exists, determine the general solution.

x+y+3 z=5

2x-y+4 z=11

-y+z=3