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 $h\in H$ […]

Linear Transformation, Basis For the Range, Rank, and Nullity, Not Injective
Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients.
Let $T: P_3 \to V$ be the linear transformation defined by
\[T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix}
a_0+a_2 & […]

All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$
Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.
Solution.
Let $T:\R^2 \to \R^2$ be a linear transformation that maps the line $y=x$ to the line $y=-x$.
Note that the linear […]

If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain
Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.
Definitions: zero divisor, integral domain
An element $a$ of a commutative ring $R$ is called a zero divisor […]

Characteristic Polynomials of $AB$ and $BA$ are the Same
Let $A$ and $B$ be $n \times n$ matrices.
Prove that the characteristic polynomials for the matrices $AB$ and $BA$ are the same.
Hint.
Consider the case when the matrix $A$ is invertible.
Even if $A$ is not invertible, note that $A-\epsilon I$ is invertible matrix […]