Describe the possible echelon forms for matrices with the following properties
$a) A$ is a $2 \times 2$ matrix with linearly dependent columns
$b) A$ is a $4 \times 3$ matrix, $A = [\vec{a_1} \vec{a_2} \vec{a_3}]$ such that ${\{ \vec{a_1} \vec{a_2}\} }$ is linearly independent and $\vec{a_3}$ is not in the span of $\vec{a_1}$ and $\vec{a_2}$
I'm not really sure how to answer these. What I'm thinking is that for $a)$– a $2 \times 2$ matrix with linearly dependent columns could mean that all of the pivots = 0 or that the $det(A) = 0$ ? I'm not quite sure.
for $b)$ – I honestly have no clue , any help would be great
UPDATE:
$a)$ The columns of the 2×2 matrix could $\textbf{not}$ have the form $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ because as long as it has only the trivial solution and 2 basic variables so would mean that linearly independent so for it to be linearly dependent it would need to be a matrix of all 0's or a matrix of at least one free variable?
Best Answer
$\textbf{Part a)}$ Two vectors are linearly dependent if and only if one is a scalar multiple of the other.
So in a)
$$A = \begin{pmatrix} a & ca \\ b & cb \end{pmatrix}$$
There are some cases: If both $a,b=0$, then $A$ is the zero matrix and that is its reduced row echelon.
If $a\neq 0$ or $b\neq 0$, then the reduced row echelon form of this matrix is:
$$\begin{pmatrix} 1 & c \\ 0 & 0 \\ \end{pmatrix}_.$$
$\textbf{Part b)}$ The conditions give that $a_1,a_2,a_3$ are linearly independent. Therefore the rank of the matrix is $3$ and the rref is:
$$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}_.$$