# Tagged: rank

## Problem 154

Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right )=\begin{bmatrix} x_1-x_2 \\ x_1+x_2 \\ x_2 \end{bmatrix}$.

(a) Show that $T$ is a linear transformation.

(b) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

(c) Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

## Problem 140

Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.
The dimension of the nullspace of $A$ is called the nullity of $A$.
Prove the followings.

(a) $\calN(A)=\calN(A^{\trans}A)$.

(b) $\rk(A)=\rk(A^{\trans}A)$.

## Problem 136

Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

## Problem 135

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.

(a) $\rk(AB) \leq \rk(A)$.

(b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.

## Problem 103

Find the rank of the following real matrix.
$\begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},$ where $a$ is a real number.

(Kyoto University, Linear Algebra Exam)

## Problem 101

For which choice(s) of the constant $k$ is the following matrix invertible?
$A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &k \\ 1 & 4 & k^2 \end{bmatrix}.$
(Johns Hopkins University, Linear Algebra Exam)

## Problem 60

Let $T: \R^3 \to \R^3$ be the linear transformation given by orthogonal projection to the line spanned by $\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}$.

(a) Find a formula for $T(\mathbf{x})$ for $\mathbf{x}\in \R^3$.

(b) Find a basis for the image subspace of $T$.

(c) Find a basis for the kernel subspace of $T$.

(d) Find the $3 \times 3$ matrix for $T$ with respect to the standard basis for $\R^3$.

(e) Find a basis for the orthogonal complement of the kernel of $T$. (The orthogonal complement is the subspace of all vectors perpendicular to a given subspace, in this case, the kernel.)

(f) Find a basis for the orthogonal complement of the image of $T$.

(g) What is the rank of $T$?

(Johns Hopkins University Exam)