# Tagged: Johns Hopkins

## Problem 180

Suppose the following information is known about a $3\times 3$ matrix $A$.
$A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=6\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}=3\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix}=3\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}.$

(a) Find the eigenvalues of $A$.

(b) Find the corresponding eigenspaces.

(c) In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.
Is $A$ a diagonalizable matrix?
Is $A$ an invertible matrix?
Is $A$ an idempotent matrix?

(Johns Hopkins 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)