Dot Product, Lengths, and Distances of Complex Vectors
For this problem, use the complex vectors
\[ \mathbf{w}_1 = \begin{bmatrix} 1 + i \\ 1 - i \\ 0 \end{bmatrix} , \, \mathbf{w}_2 = \begin{bmatrix} -i \\ 0 \\ 2 - i \end{bmatrix} , \, \mathbf{w}_3 = \begin{bmatrix} 2+i \\ 1 - 3i \\ 2i \end{bmatrix} . \]
Suppose $\mathbf{w}_4$ is […]

If Two Matrices Have the Same Rank, Are They Row-Equivalent?
If $A, B$ have the same rank, can we conclude that they are row-equivalent?
If so, then prove it. If not, then provide a counterexample.
Solution.
Having the same rank does not mean they are row-equivalent.
For a simple counterexample, consider $A = […]

Inner Products, Lengths, and Distances of 3-Dimensional Real Vectors
For this problem, use the real vectors
\[ \mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} . \]
Suppose that $\mathbf{v}_4$ is another vector which is […]

Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible?
A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix.
Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.
Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of […]

The Length of a Vector is Zero if and only if the Vector is the Zero Vector
Let $\mathbf{v}$ be an $n \times 1$ column vector.
Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.
Proof.
Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} $.
Then we […]

A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$
An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that
$AB=I$, and
$BA=I$,
where $I$ is the $n\times n$ identity matrix.
If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted […]

7 Problems on Skew-Symmetric Matrices
Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$.
(a) Prove that $A+B$ is skew-symmetric.
(b) Prove that $cA$ is skew-symmetric for any scalar $c$.
(c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]

Any Automorphism of the Field of Real Numbers Must be the Identity Map
Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.
Proof.
We prove the problem by proving the following sequence of claims.
Let $\phi:\R \to \R$ be an automorphism of the field of real numbers […]