Let
\[A=\begin{bmatrix}
a & -1\\
1& 4
\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is some real number.
Suppose that the matrix $A$ has an eigenvalue $3$.
(a) Determine the value of $a$.
(b) Does the matrix $A$ have eigenvalues other than $3$?
Add to solve later Suppose that $\lambda$ and $\mu$ are two distinct eigenvalues of a square matrix $A$ and let $\mathbf{x}$ and $\mathbf{y}$ be eigenvectors corresponding to $\lambda$ and $\mu$, respectively.
If $a$ and $b$ are nonzero numbers, then prove that $a \mathbf{x}+b\mathbf{y}$ is not an eigenvector of $A$ (corresponding to any eigenvalue of $A$).
Add to solve later Use Cramer’s rule to solve the system of linear equations
\begin{align*}
3x_1-2x_2&=5\\
7x_1+4x_2&=-1.
\end{align*}
Read solution
Add to solve later Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients.
Let $W$ be the subspace of $P_2$ by
\[W=\{ p(x)\in P_4 \mid p(1)+p(-1)=0 \text{ and } p(2)+p(-2)=0 \}.\]
Find a basis of the subspace $W$ and determine the dimension of $W$.
Add to solve later Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that
\[T(\mathbf{v}_1)=\begin{bmatrix}
1 \\
-2
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
3 \\
1
\end{bmatrix}.\]
If $\mathbf{e}_1=\mathbf{v}_1+2\mathbf{v}_2 \text{ and } \mathbf{e}_2=2\mathbf{v}_1-\mathbf{u}_2$, where $\mathbf{e}_1, \mathbf{e}_2$ are the standard unit vectors in $\R^2$, then find the matrix of $T$ with respect to the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$.
Read solution
Add to solve later Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are
\[\|\mathbf{a}\|=\|\mathbf{b}\|=1\]
and the inner product
\[\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.\]
Then determine the length $\|\mathbf{a}-\mathbf{b}\|$.
(Note that this length is the distance between $\mathbf{a}$ and $\mathbf{b}$.)
Add to solve later Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample.
Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$.
If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a basis of the subspace $W_1\cap W_2$.
Add to solve later Let $W$ be the subset of $\R^3$ defined by
\[W=\left \{ \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}\in \R^3 \quad \middle| \quad 5x_1-2x_2+x_3=0 \right \}.\]
Exhibit a $1\times 3$ matrix $A$ such that $W=\calN(A)$, the null space of $A$.
Conclude that the subset $W$ is a subspace of $\R^3$.
Add to solve later Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.
Add to solve later Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies
\[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\]
Define the matrix
\[A=I+\mathbf{u}\mathbf{v}^{\trans}.\]
Prove that $A$ is invertible and the inverse matrix is given by the formula
\[A^{-1}=I-a\mathbf{u}\mathbf{v}^{\trans},\]
where
\[a=\frac{1}{1+\mathbf{v}^{\trans}\mathbf{u}}.\]
This formula is called the Sherman-Woodberry formula.
Add to solve later Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.
\[A= \left[\begin{array}{rrr|r}
1 & 2 & 3 & 4 \\
2 &-1 & -2 & a^2 \\
-1 & -7 & -11 & a
\end{array} \right],\]
where $a$ is a real number. Determine all the values of $a$ so that the corresponding system is consistent.
Add to solve later We say that two $m\times n$ matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations.
Let $A$ and $I$ be $2\times 2$ matrices defined as follows.
\[A=\begin{bmatrix}
1 & b\\
c& d
\end{bmatrix}, \qquad I=\begin{bmatrix}
1 & 0\\
0& 1
\end{bmatrix}.\]
Prove that the matrix $A$ is row equivalent to the matrix $I$ if $d-cb \neq 0$.
Read solution
Add to solve later Let $R$ be a commutative ring with unity. A proper ideal $I$ of $R$ is called primary if whenever $ab \in I$ for $a, b\in R$, then either $a\in I$ or $b^n\in I$ for some positive integer $n$.
(a) Prove that a prime ideal $P$ of $R$ is primary.
(b) If $P$ is a prime ideal and $a^n\in P$ for some $a\in R$ and a positive integer $n$, then show that $a\in P$.
(c) If $P$ is a prime ideal, prove that $\sqrt{P}=P$.
(d) If $Q$ is a primary ideal, prove that the radical ideal $\sqrt{Q}$ is a prime ideal.
Add to solve later Let $H$ be a subgroup of a group $G$. We call $H$ characteristic in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$.
(a) Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$.
(b) Prove that the center $Z(G)$ of $G$ is characteristic in $G$.
Add to solve later Let $p, q$ be prime numbers such that $p>q$.
If a group $G$ has order $pq$, then show the followings.
(a) The group $G$ has a normal Sylow $p$-subgroup.
(b) The group $G$ is solvable.
Add to solve later Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms.
Define the subset $M$ of $G_1 \times G_2$ to be
\[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\]
Prove that $M$ is a subgroup of $G_1 \times G_2$.
Add to solve later Let $f:G\to G’$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$.
Then prove that a group homomorphism $f: G \to G’$ is injective if and only if it is monic.
Add to solve later Let
\[A=\begin{bmatrix}
1 & 2 & 2 \\
2 &3 &2 \\
-1 & -3 & -4
\end{bmatrix} \text{ and }
B=\begin{bmatrix}
1 & 2 & 2 \\
2 &3 &2 \\
5 & 3 & 3
\end{bmatrix}.\]
Determine the null spaces of matrices $A$ and $B$.
Add to solve later A nontrivial abelian group $A$ is called divisible if for each element $a\in A$ and each nonzero integer $k$, there is an element $x \in A$ such that $x^k=a$.
(Here the group operation of $A$ is written multiplicatively. In additive notation, the equation is written as $kx=a$.) That is, $A$ is divisible if each element has a $k$-th root in $A$.
(a) Prove that the additive group of rational numbers $\Q$ is divisible.
(b) Prove that no finite abelian group is divisible.
Add to solve later 
