Tagged: exam

Express a Vector as a Linear Combination of Other Vectors


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)

 
Read solution

LoadingAdd to solve later

10 True or False Problems about Basic Matrix Operations

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.

 
Read solution

LoadingAdd to solve later

Possibilities For the Number of Solutions for a Linear System

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)
Read solution

LoadingAdd to solve later

True or False Quiz About a System of Linear Equations

Problem 78

Determine whether the following sentence is True or False.

(Purdue University Linear Algebra Exam)

Read solution

LoadingAdd to solve later

A Linear Transformation from Vector Space over Rational Numbers to itself

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)

Read solution

LoadingAdd to solve later

Solving a System of Linear Equations By Using an Inverse Matrix

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)

Read solution

LoadingAdd to solve later

A Square Root Matrix of a Symmetric Matrix

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)

Read solution

LoadingAdd to solve later

Find All Matrices Satisfying a Given Relation

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)

Read solution

LoadingAdd to solve later

Find a Basis of the Subspace of All Vectors that are Perpendicular to the Columns of the Matrix

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)

Read solution

LoadingAdd to solve later

Given the Characteristic Polynomial of a Diagonalizable Matrix, Find the Size of the Matrix, Dimension of Eigenspace

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)

Read solution

LoadingAdd to solve later

Find a Formula for a Linear Transformation

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)

Read solution

LoadingAdd to solve later