Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.
(a) $\rk(AB) \leq \rk(A)$.
(b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.
Add to solve later Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not.
(a) $S=\{f(x) \in V \mid f(0)=f(1)\}$.
(b) $T=\{f(x) \in V \mid f(0)=f(1)+3\}$.
Add to solve later Let
\[A=\begin{bmatrix}
1 & 1 & 0 \\
1 &1 &0
\end{bmatrix}\]
be a matrix.
Find a basis of the null space of the matrix $A$.
(Remark: a null space is also called a kernel.)
Add to solve later Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$.
\[V:=\left\{ \quad\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \in \R^4
\quad \middle| \quad
x_1-x_2+x_3-x_4=0 \quad\right\}.\]
Find a basis of the subspace $V$ and its dimension.
Add to solve later Let $S$ be the following subset of the 3-dimensional vector space $\R^3$.
\[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \]
where $\Z$ is the set of all integers.
Determine whether $S$ is a subspace of $\R^3$.
Add to solve later Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by
\[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\]
That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$.
Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$.
(Note that the null space is also called the kernel of $A$.)
Read solution
Add to solve later Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by
\[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\]
Prove that the subset $W$ is a subspace of $\R^3$.
Read solution
Add to solve later Let $V$ be a finite dimensional vector space over a field $K$ and let $\End (V)$ be the vector space of linear transformations from $V$ to $V$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ be a basis for $V$.
Show that the map $\phi:\End (V) \to V^{\oplus n}$ defined by $f\mapsto (f(\mathbf{v}_1), \dots, f(\mathbf{v}_n))$ is an isomorphism.
Here $V^{\oplus n}=V\oplus \dots \oplus V$, the direct sum of $n$ copies of $V$.
Read solution
Add to solve later Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.
That is,
\begin{equation*}
V:=\left\{ A=\begin{bmatrix}
a_{11} & 0 & \dots & 0 \\
0 &a_{22} & \dots & 0 \\
0 & 0 & \ddots & \vdots \\
0 & 0 & \dots & a_{nn}
\end{bmatrix} \quad \middle| \quad
\begin{array}{l}
a_{11}, \dots, a_{nn} \in \C,\\
\tr(A)=0 \\
\end{array}
\right\}
\end{equation*}
Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.
(a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)
(b) Show that matrices
\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.
(c) Find the dimension of $V$.
Read solution
Add to solve later Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less.
Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$.
For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation of $T$ with respect to the basis $B$. For $f(x)\in P_2(\R)$, define $T$ as follows.
(a) \[T(f(x))=\frac{\mathrm{d}^2}{\mathrm{d}x^2} f(x)-3\frac{\mathrm{d}}{\mathrm{d}x}f(x)\]
(b) \[T(f(x))=\int_{-1}^1\! (t-x)^2f(t) \,\mathrm{d}t\]
(c) \[T(f(x))=e^x \frac{\mathrm{d}}{\mathrm{d}x}(e^{-x}f(x))\]
Add to solve later Let $V$ and $W$ be subspaces of $\R^n$ such that $V \cap W =\{\mathbf{0}\}$ and $\dim(V)+\dim(W)=n$.
(a) If $\mathbf{v}+\mathbf{w}=\mathbf{0}$, where $\mathbf{v}\in V$ and $\mathbf{w}\in W$, then show that $\mathbf{v}=\mathbf{0}$ and $\mathbf{w}=\mathbf{0}$.
(b) If $B_1$ is a basis for the subspace $V$ and $B_2$ is a basis for the subspace $W$, then show that the union $B_1\cup B_2$ is a basis for $R^n$.
(c) If $\mathbf{x}$ is in $\R^n$, then show that $\mathbf{x}$ can be written in the form $\mathbf{x}=\mathbf{v}+\mathbf{w}$, where $\mathbf{v}\in V$ and $\mathbf{w} \in W$.
(d) Show that the representation obtained in part (c) is unique.
Add to solve later 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)
Add to solve later Let $V$ be an $n$-dimensional vector space over a field $K$.
Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$.
Are the following vectors linearly independent?
\[\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, \quad \dots, \quad \mathbf{v}_{k-1}+\mathbf{v}_k, \quad \mathbf{v}_k+\mathbf{v}_1.\]
If it is linearly dependent, give a non-trivial linear combination of these vectors summing up to the zero vector.
Add to solve later 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)
Add to solve later Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space.
(MIT-Massachusetts Institute of Technology Exam)
Add to solve later Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent?
(a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ for $i=1,2,3,4$.
(b) At $0$ each of the polynomials has the value $1$. Namely $p_i(0)=1$ for $i=1,2,3,4$.
(University of California, Berkeley)
Add to solve later 
