# Tagged: exam

## Problem 115

Express the vector $\mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 \end{bmatrix}$ as a linear combination of the vectors
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2= \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{v}_3= \begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}.$

(The Ohio State University, Linear Algebra Exam)

## Problem 114

Let
$A=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix} 1\\ 0 \end{bmatrix}.$ Compute $A^{2017}\mathbf{u}$.

(The Ohio State University, Linear Algebra Exam)

## Problem 104

Test your understanding of basic properties of matrix operations.

There are 10 True or False Quiz Problems.

These 10 problems are very common and essential.
So make sure to understand these and don’t lose a point if any of these is your exam problems.
(These are actual exam problems at the Ohio State University.)

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.

## 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 102

Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.

(a) $\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right.$ where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.

(b) $A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix.

(c) A homogeneous system of $3$ equations in $4$ unknowns.

(d) $A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:
$\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$ (The Ohio State 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 93

4 multiple choice questions about possibilities for the solution set of a homogeneous system of linear equations.

The solutions will be given after completing all problems.

## Problem 78

Determine whether the following sentence is True or False.

(Purdue University Linear Algebra Exam)

## Problem 76

Let A be the matrix
$\begin{bmatrix} 1 & -1 & 0 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}.$ Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.

(The Ohio State University Linear Algebra Exam)

## Problem 75

Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.

(a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the vector space $V$ over $\Q$.

(b) Let $\alpha=a+b\sqrt{2} \in V$, and let $T_{\alpha}: V \to V$ be the map defined by
$T_{\alpha}(x+y\sqrt{2}):=(ax+2by)+(ay+bx)\sqrt{2}\in V$ for any $x+y\sqrt{2} \in V$.
Show that $T_{\alpha}$ is a linear transformation.

(c) Let $\begin{bmatrix} x \\ y \end{bmatrix}_B=x+y \sqrt{2}$.
Find the matrix $T_B$ such that
$T_{\alpha} (x+y \sqrt{2})=\left( T_B\begin{bmatrix} x \\ y \end{bmatrix}\right)_B,$ and compute $\det T_B$.

(The Ohio State University, Linear Algebra Exam)

## Problem 65

Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}

(a) Find the coefficient matrix and its inverse matrix.

(b) Using the inverse matrix, solve the system of linear equations.

(The Ohio State University, Linear Algebra Exam)

## Problem 59

Answer the following two questions with justification.

(a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix.

(b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ where
$A=\begin{bmatrix} 1 & -1 & 0 \\ -1 &2 &-1 \\ 0 & -1 & 1 \end{bmatrix}\,\,\,\,?$

(Princeton University Linear Algebra Exam)

## Problem 55

Let $A$ and $B$ are $n \times n$ matrices with real entries.
Assume that $A+B$ is invertible. Then show that
$A(A+B)^{-1}B=B(A+B)^{-1}A.$

(University of California, Berkeley Qualifying Exam)

## Problem 47

Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$.
Calculate and simplify the expression
$-T^3+4T^2+5T-2I,$ where $I$ is the $3\times 3$ identity matrix.

(The Ohio State University Linear Algebra Exam)

## Problem 43

Let $a$ and $b$ be two distinct positive real numbers. Define matrices
$A:=\begin{bmatrix} 0 & a\\ a & 0 \end{bmatrix}, \,\, B:=\begin{bmatrix} 0 & b\\ b& 0 \end{bmatrix}.$

Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a non-zero real matrix satisfying the relation
$AX+XB=\lambda X. \tag{*}$

(The University of Tokyo Linear Algebra Exam)

## Problem 41

Find $A^{10}$, where $A=\begin{bmatrix} 4 & 3 & 0 & 0 \\ 3 &-4 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}$.

(Harvard University Exam)

## Problem 40

Find a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix
$A=\begin{bmatrix} 11 & 12 & 13 & 14 \\ 21 &22 & 23 & 24 \\ 31 & 32 & 33 & 34 \\ 41 & 42 & 43 & 44 \end{bmatrix}.$

(Harvard University Exam)

## Problem 39

Suppose that $A$ is a diagonalizable matrix with characteristic polynomial
$f_A(\lambda)=\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3.$

(a) Find the size of the matrix $A$.

(b) Find the dimension of $E_4$, the eigenspace corresponding to the eigenvalue $\lambda=4$.

(c) Find the dimension of the kernel(nullspace) of $A$.

(Stanford University Linear Algebra Exam)

## Problem 37

Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues.
Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.

(Stanford University Linear Algebra Exam)

See below for a generalized problem.

## Problem 36

If $L:\R^2 \to \R^3$ is a linear transformation such that
\begin{align*}
L\left( \begin{bmatrix}
1 \\
0
\end{bmatrix}\right)
=\begin{bmatrix}
1 \\
1 \\
2
\end{bmatrix}, \,\,\,\,
L\left( \begin{bmatrix}
1 \\
1
\end{bmatrix}\right)
=\begin{bmatrix}
2 \\
3 \\
2
\end{bmatrix}.
\end{align*}
then

(a) find $L\left( \begin{bmatrix} 1 \\ 2 \end{bmatrix}\right)$, and

(b) find the formula for $L\left( \begin{bmatrix} x \\ y \end{bmatrix}\right)$.

If you think you can solve (b), then skip (a) and solve (b) first and use the result of (b) to answer (a).

(Part (a) is an exam problem of Purdue University)