Let $T: \R^n \to \R^m$ be a linear transformation.
Suppose that the nullity of $T$ is zero.
If $\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then show that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$.
Add to solve later Find the vector form solution $\mathbf{x}$ of the equation $A\mathbf{x}=\mathbf{0}$, where $A=\begin{bmatrix}
1 & 1 & 1 & 1 &2 \\
1 & 2 & 4 & 0 & 5 \\
3 & 2 & 0 & 5 & 2 \\
\end{bmatrix}$. Also, find two linearly independent vectors $\mathbf{x}$ satisfying $A\mathbf{x}=\mathbf{0}$.
Add to solve later Let $A$ be an $n\times n$ nonsingular matrix. Let $\mathbf{v}, \mathbf{w}$ be linearly independent vectors in $\R^n$. Prove that the vectors $A\mathbf{v}$ and $A\mathbf{w}$ are linearly independent.
Add to solve later Let $\mathbf{P}_2$ be the vector space of polynomials of degree $2$ or less.
(a) Prove that the set $\{ 1 , 1 + x , (1 + x)^2 \}$ is a basis for $\mathbf{P}_2$.
(b) Write the polynomial $f(x) = 2 + 3x – x^2$ as a linear combination of the basis $\{ 1 , 1+x , (1+x)^2 \}$.
Add to solve later Suppose $M$ is an $n \times n$ upper-triangular matrix.
If the diagonal entries of $M$ are all non-zero, then prove that the column vectors are linearly independent.
Does the conclusion hold if we do not assume that $M$ has non-zero diagonal entries?
Add to solve later Prove that any set of vectors which contains the zero vector is linearly dependent.
Add to solve later Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.
(a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.
(b) Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$.
Add to solve later 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 independent.
(The Ohio State University, Linear Algebra Midterm)
Read solution
Add to solve later Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$.
Suppose that $S$ is an orthogonal set.
(a) Show that $S$ is linearly independent.
(b) If $k=n$, then prove that $S$ is a basis for $\R^n$.
Add to solve later Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let
\[V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}\]
be a subset in $C[-1, 1]$.
(a) Prove that $V$ is a subspace of $C[-1, 1]$.
(b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.
(c) Prove that
\[B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}\]
is a basis for $V$.
Add to solve later Determine whether each of the following sets is a basis for $\R^3$.
(a) $S=\left\{\, \begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix}, \begin{bmatrix}
2 \\
1 \\
-1
\end{bmatrix}, \begin{bmatrix}
-2 \\
1 \\
4
\end{bmatrix} \,\right\}$
(b) $S=\left\{\, \begin{bmatrix}
1 \\
4 \\
7
\end{bmatrix}, \begin{bmatrix}
2 \\
5 \\
8
\end{bmatrix}, \begin{bmatrix}
3 \\
6 \\
9
\end{bmatrix} \,\right\}$
(c) $S=\left\{\, \begin{bmatrix}
1 \\
1 \\
2
\end{bmatrix}, \begin{bmatrix}
0 \\
1 \\
7
\end{bmatrix} \,\right\}$
(d) $S=\left\{\, \begin{bmatrix}
1 \\
2 \\
5
\end{bmatrix}, \begin{bmatrix}
7 \\
4 \\
0
\end{bmatrix}, \begin{bmatrix}
3 \\
8 \\
6
\end{bmatrix}, \begin{bmatrix}
-1 \\
9 \\
10
\end{bmatrix} \,\right\}$
Add to solve later Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where
\[\mathbf{a}=\begin{bmatrix}
1 \\
0 \\
1 \\
0
\end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix}
1 \\
1 \\
0 \\
0
\end{bmatrix}, \quad \mathbf{c}=\begin{bmatrix}
0 \\
1 \\
-1 \\
0
\end{bmatrix}.\]
Namely,
\[V=\{\mathbf{x}\in \R^4 \mid \mathbf{a}^{\trans}\mathbf{x}=0, \mathbf{b}^{\trans}\mathbf{x}=0, \text{ and } \mathbf{c}^{\trans}\mathbf{x}=0\}.\]
(a) Prove that $V$ is a subspace of $\R^4$.
(b) Find a basis of $V$.
(c) Determine the dimension of $V$.
Add to solve later Let $V$ be a subspace of $\R^n$.
Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$.
Prove that every basis of $V$ consists of $k$ vectors in $V$.
Add to solve later Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.
(a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.
(b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of $\R^3$.
Add to solve later Let
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
1 \\
a \\
5
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
0 \\
4 \\
b
\end{bmatrix}\]
be vectors in $\R^3$.
Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.
Add to solve later Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector.
Then the product $A\mathbf{b}$ is an $n$-dimensional vector.
Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$.
Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$.
Add to solve later By calculating the Wronskian, determine whether the set of exponential functions
\[\{e^x, e^{2x}, e^{3x}\}\]
is linearly independent on the interval $[-1, 1]$.
Add to solve later 10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.
The quiz is designed to test your understanding of the basic properties of these topics.
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.
Add to solve later Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation).
Let $V$ be the vector space of all linear transformations from $W$ to $W$.
The addition and the scalar multiplication of $V$ are given by those of linear transformations.
Let $T_1, T_2, T_3$ be the elements in $V$ defined by
\begin{align*}
T_1\left(\, f(x) \,\right)&=\frac{\mathrm{d}}{\mathrm{d}x}f(x)\\[6pt]
T_2\left(\, f(x) \,\right)&=\frac{\mathrm{d}^2}{\mathrm{d}x^2}f(x)\\[6pt]
T_3\left(\, f(x) \,\right)&=\int_{0}^x \! f(t)\,\mathrm{d}t.
\end{align*}
Then determine whether the set $\{T_1, T_2, T_3\}$ are linearly independent or linearly dependent.
Add to solve later Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
\[\left(\, f\left(\, \begin{bmatrix}
\alpha \\
\beta
\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.\]
We put
\[V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.\]
(a) Prove that the map $f$ is a linear transformation.
(b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.
(c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
(This yields an isomorphism of $\R^2$ and $V$.)
(d) Define a map $g:V \to V$ by
\[g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.\]
Prove that the map $g$ is a linear transformation.
(e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.
(Kyoto University, Linear Algebra exam problem)
Add to solve later 
