Problem Set 6: Rank, Nullity, and Bases

Let \[ A = \begin{bmatrix*}[r] 1 & 2 & 0 & 1 & 3 \\ 2 & 4 & 1 & 3 & 5 \\ 1 & 2 & 1 & 2 & 4 \\ 0 & 0 & 1 & 1 & -1 \end{bmatrix*}. \] You may use the Row Reducer to help you calculate.

1. Find $\rank(A)$ and $\nullity(A)$.
2. Is the equation $Ax = b$ consistent for every vector $b$ in $\RR^4$? Explain.
3. Let \[ w_1 = \begin{bmatrix*}[r] 1 \\ 0 \\ 1 \\ -1 \\ 0 \end{bmatrix*},\qquad w_2 = \begin{bmatrix*}[r] -1 \\ 1 \\ 1 \\ -1 \\ 0 \end{bmatrix*}. \] Explain why $w_1, w_2$ is a basis of $\Nul(A)$.

1. $A$ is a $3 \times 7$ matrix, and the solution set of $Ax = 0$ is a plane through the origin. What is $\rank(A)$?
2. $A$ is a $6 \times 4$ matrix, and for some vector $b$ the equation $Ax = b$ has exactly one solution. What is $\rank(A)$?
3. $A$ is a $3 \times 5$ matrix, and $\col(A)$ is a line through the origin. How many free variables does the equation $Ax = 0$ have?
4. $A$ is a $3 \times 5$ matrix, explain why the solution set of $Ax = 0$ cannot be a line through the origin.

In this problem we will play a game called Name That Matrix! In each part you are given a description of a linear transformation $T_A : \RR^2 \to \RR^2$, and your job is to find the matrix $A$.

1. Find the matrix $A$ such that $T_A(e_1) = (3,4)$ and $T_A(e_2) = (-1,0)$.

2. Find the matrix $A$ such that $T_A$ maps the face on the left to the face on the right.
3. Find the matrix $A$ such that $T_A$ doubles the length of every vector and rotates it $180^\circ$ around the origin.
4. Find the matrix $A$ such that $T_A(1, 1) = (8, 1)$ and $T_A(1, -1) = (2, -3)$.
5. Find the matrix $A$ such that $T_A$ reflects the plane across the line consisting of the points $(3t, 4t)$.

6. Find the matrix $A$ such that $T_A$ maps the face on the left to the face on the right.

Let $P$ be the plane in $\RR^3$ with equation $x + y + z = 0$. The vectors \[ v_1 = \begin{bmatrix*}[r] 1 \\ -1 \\ 0 \end{bmatrix*},\qquad v_2 = \begin{bmatrix*}[r] 0 \\ 1 \\ -1 \end{bmatrix*} \] lie in $P$, and the vector $w = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ is perpendicular to $P$. Let $\cB = \{v_1, v_2, w\}$.

1. Explain why $\cB$ is a basis of $\RR^3$.
2. Let $u = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$. Find $[u]_\cB$.
3. Let $u'$ be the reflection of $u$ across the plane $P$. Note that reflecting in $P$ leaves every vector in $P$ where it is and sends $w$ to $-w$. Find $[u']_\cB$.
4. Find $u'$.
5. Let $L$ be the line through the origin in the direction of $w$. Rotating $\RR^3$ halfway around $L$ sends $w$ to itself and sends every vector in $P$ to its negative. Find the vector $z$ obtained by rotating $u$ halfway around $L$.