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 $M \mathbf{x} = \mathbf{v}$.
Add to solve later Let $A$ be a singular $n\times n$ matrix.
Let
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0 \\
\vdots \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1 \\
\vdots \\
0
\end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix}
0 \\
0 \\
\vdots \\
1
\end{bmatrix}\]
be unit vectors in $\R^n$.
Prove that at least one of the following matrix equations
\[A\mathbf{x}=\mathbf{e}_i\]
for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.
Add to solve later 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 $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.
(b) $A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix.
(c) A homogeneous system of $3$ equations in $4$ unknowns.
(d) $A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:
\[\begin{bmatrix}
1 & 0 & -1 & 0 \\
0 &1 & 2 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.\]
(The Ohio State University, Linear Algebra Exam)
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