Find a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix
\[A=\begin{bmatrix}
11 & 12 & 13 & 14 \\
21 &22 & 23 & 24 \\
31 & 32 & 33 & 34 \\
41 & 42 & 43 & 44
\end{bmatrix}.\]

Find a basis of $\ker(A^{\trans})$ by reducing the matrix $A^{\trans}$.

The kernel of $A$ is also called the null space of $A$ and it is denoted by $\calN(A)$.
So $\ker(A)=\calN(A)$.

Solution.

Let us write $A=[A_1 \, A_2 \, A_3 \, A_4]$, where $A_i$ is the $i$-th column vector of $A$ for $i=1,2,3,4$.
First we claim that a vector $x\in \R^4$ is perpendicular to all column vectors $A_i$ if and only if $x\in \ker(A^{\trans})$.
To see this, we compute
\begin{align*}
A^{\trans}x=\begin{bmatrix}
A_1^{\trans} \\
A_2^{\trans} \\
A_3^{\trans} \\
A_4^{\trans}
\end{bmatrix}x
=\begin{bmatrix}
A_1^{\trans}x \\
A_2^{\trans} x\\
A_3^{\trans} x\\
A_4^{\trans} x
\end{bmatrix}.
\end{align*}
From this equality the claim follows immediately.

So we proved that $\ker(A^{\trans}) =W$. From this, we see that $W$ is actually a subspace in $\R^4$.
Thus, we need to find a basis for the kernel of the transpose $A^{\trans}$.

We apply elementary row operations to $A^{\trans}$ and obtain a reduced row echelon form
\[A^{\trans} \to \begin{bmatrix}
1 & 0 & -1 & -2 \\
0 &1 & 2 & 3 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}.\]
The last two columns correspond to two free variables. Let $s$ and $t$ be free variable.
Then $x=\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \in \ker(A^{\trans})$ if and only if $x$ satisfies
\begin{align*}
x_1 &=s+2t \\
x_2 &=-2s-3t\\
x_3 &=s\\
x_4 &=t,
\end{align*}
equivalently
\begin{align*}
x=s\begin{bmatrix}
1 \\
-2 \\
1 \\
0
\end{bmatrix}
+t\begin{bmatrix}
2 \\
-3 \\
0 \\
1
\end{bmatrix}.
\end{align*}
Therefore a basis of $W=\ker(A^{\trans})$ is
\[ \begin{bmatrix}
1 \\
-2 \\
1 \\
0
\end{bmatrix} \text{ and } \begin{bmatrix}
2 \\
-3 \\
0 \\
1
\end{bmatrix}.\]

Find a Basis for a Subspace of the Vector Space of $2\times 2$ Matrices
Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where
\begin{align*}
A_1=\begin{bmatrix}
1 & 2 \\
-1 & 3
\end{bmatrix}, \quad
A_2=\begin{bmatrix}
0 & -1 \\
1 & 4
[…]

Calculate $A^{10}$ for a Given Matrix $A$
Find $A^{10}$, where $A=\begin{bmatrix}
4 & 3 & 0 & 0 \\
3 &-4 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1
\end{bmatrix}$.
(Harvard University exam)
Solution.
Let $B=\begin{bmatrix}
4 & 3\\
3& -4
\end{bmatrix}$ […]

Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.
Hint.
Recall that the rank of a matrix $A$ is the dimension of the range of $A$.
The range of $A$ is spanned by the column vectors of the matrix […]

Subspace of Skew-Symmetric Matrices and Its Dimension
Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.)
(a) Prove that the subset $W$ is a subspace of $V$.
(b) Find the […]

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 $a,b,c, d$ […]

Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)
(a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.
Find $\det(A)$.
(b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.
(c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of […]

Determinant of Matrix whose Diagonal Entries are 6 and 2 Elsewhere
Find the determinant of the following matrix
\[A=\begin{bmatrix}
6 & 2 & 2 & 2 &2 \\
2 & 6 & 2 & 2 & 2 \\
2 & 2 & 6 & 2 & 2 \\
2 & 2 & 2 & 6 & 2 \\
2 & 2 & 2 & 2 & 6
\end{bmatrix}.\]
(Harvard University, Linear Algebra Exam […]

Suppose that $A$ is a diagonalizable matrix with characteristic polynomial \[f_A(\lambda)=\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3.\] (a) Find the size of the matrix $A$. (b) Find the...