Group Homomorphism, Preimage, and Product of Groups
Let $G, G'$ be groups and let $f:G \to G'$ be a group homomorphism.
Put $N=\ker(f)$. Then show that we have
\[f^{-1}(f(H))=HN.\]
Proof.
$(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$.
It follows that there exists $h\in H$ […]

12 Examples of Subsets that Are Not Subspaces of Vector Spaces
Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) \[S_1=\left \{\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in […]

The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive
Let $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers.
Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive?
If so, prove it. Otherwise, give a counterexample.
Solution.
The […]

Complement of Independent Events are Independent
Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$.
Prove that $E$ and $F^c$ are independent as well.
Solution.
Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that
\[P(E) = P(E \cap F) + P(E […]

Diagonalize a 2 by 2 Matrix if Diagonalizable
Determine whether the matrix
\[A=\begin{bmatrix}
1 & 4\\
2 & 3
\end{bmatrix}\]
is diagonalizable.
If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
(The Ohio State University, Linear Algebra Final Exam […]

Stochastic Matrix (Markov Matrix) and its Eigenvalues and Eigenvectors
(a) Let
\[A=\begin{bmatrix}
a_{11} & a_{12}\\
a_{21}& a_{22}
\end{bmatrix}\]
be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$.
(Such a matrix is called (right) stochastic matrix (also termed […]

Possibilities For the Number of Solutions for a Linear System
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 $a,b,c, d$ […]

Projection to the subspace spanned by a vector
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 […]