Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector
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)
Solution.
We first compute $A\mathbf{u}$. We […]

Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less
Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less.
Let
\[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\]
where
\begin{align*}
p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\
p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3.
\end{align*}
(a) […]

Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors
Let $V$ be a vector space and $B$ be a basis for $V$.
Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.
Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […]

Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix
Let
\[A=\begin{bmatrix}
1 & -1 & 0 & 0 \\
0 &1 & 1 & 1 \\
1 & -1 & 0 & 0 \\
0 & 2 & 2 & 2\\
0 & 0 & 0 & 0
\end{bmatrix}.\]
(a) Find a basis for the null space $\calN(A)$.
(b) Find a basis of the range $\calR(A)$.
(c) Find a basis of the […]

Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?
Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\]
in $C[-2\pi, 2\pi]$.
Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly […]

Find an Orthonormal Basis of the Given Two Dimensional Vector Space
Let $W$ be a subspace of $\R^4$ with a basis
\[\left\{\, \begin{bmatrix}
1 \\
0 \\
1 \\
1
\end{bmatrix}, \begin{bmatrix}
0 \\
1 \\
1 \\
1
\end{bmatrix} \,\right\}.\]
Find an orthonormal basis of $W$.
(The Ohio State […]

Vector Space of 2 by 2 Traceless Matrices
Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers.
Let
\[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix}
a & b\\
c& -a
\end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\]
(a) Show that $W$ is a subspace of […]