Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$.

Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the equation $M \mathbf{x} = \mathbf{v}$.

Suppose that $\mathbf{v}$ is a linear combination of the vectors $\mathbf{b}_1, \dots, \mathbf{v}_m$. That is, suppose there are coefficients $x_1 , x_2 , \cdots , x_m$ such that
\[x_1 \mathbf{b}_1 + x_2 \mathbf{b}_2 + \cdots + x_m \mathbf{b}_m = \mathbf{v}.\]

Then define the column vector
\[\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix}.\]
Then $\mathbf{x}$ satisfies the equation $M \mathbf{x} = \mathbf{v}$.

On the other hand, suppose that there is a vector $\mathbf{x}$ which satisfies the desired equation with components $x_1 , x_2 , \cdots, x_m$.

Then these components can be used to find the linear combination
\[x_1 \mathbf{b}_1 + x_2 \mathbf{b}_2 + \cdots + x_m \mathbf{b}_m = M \mathbf{x} = \mathbf{v}.\]
Hence, the vector $\mathbf{v}$ is a linear combination of column vectors $\mathbf{b}_1, \dots, \mathbf{b}_m$.

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) […]

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 […]

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 $a,b,c, d$ […]

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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 $\R^3$.
[…]

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(a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.
(b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.
Proof.
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Express a Vector as a Linear Combination of Other Vectors
Express the vector $\mathbf{b}=\begin{bmatrix}
2 \\
13 \\
6
\end{bmatrix}$ as a linear combination of the vectors
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
5 \\
-1
\end{bmatrix},
\mathbf{v}_2=
\begin{bmatrix}
1 \\
2 \\
1
[…]