(a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.

Note that the column vectors of the matrix $A$ are linearly dependent if the matrix equation
\[A\mathbf{x}=\mathbf{0}\]
has a nonzero solution $\mathbf{x}\in \R^5$.

The equation is equivalent to a $3\times 5$ homogeneous system.
As there are more variables than equations, the homogeneous system has infinitely many solutions.

In particular, the equation has a nonzero solution $\mathbf{x}$.
Hence the column vectors are linearly dependent.

(b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.

Observe that the row vectors of the matrix $B$ are the column vectors of the transpose $B^{\trans}$. Note that the size of $B^{\trans}$ is $3\times 5$.

In part (a), we showed that the column vectors of any $3\times 5$ matrix are linearly dependent.
It follows that the column vectors of $B^{\trans}$ are linearly dependent.
Hence the row vectors of $B$ are linearly dependent.

The Set of Vectors Perpendicular to a Given Vector is a Subspace
Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define
\[W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.\]
Prove that $W$ is a vector subspace of $\R^3$.
[…]

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 […]

Determine a Condition on $a, b$ so that Vectors are Linearly Dependent
Let
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
1 \\
a \\
5
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
0 \\
4 \\
b
\end{bmatrix}\]
be vectors in $\R^3$.
Determine a […]

If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent
Let $V$ be a subspace of $\R^n$.
Suppose that
\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\]
is a spanning set for $V$.
Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.
We give two proofs. The essential ideas behind […]

How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix
Let $A=\begin{bmatrix}
2 & 4 & 6 & 8 \\
1 &3 & 0 & 5 \\
1 & 1 & 6 & 3
\end{bmatrix}$.
(a) Find a basis for the nullspace of $A$.
(b) Find a basis for the row space of $A$.
(c) Find a basis for the range of $A$ that consists of column vectors of $A$.
(d) […]