Finding the range of a matrix

1. The problem statement, all variables and given/known data

##\begin{bmatrix}1 & 1 & -1 & -1 \\1 & 2 & 0 & 1 \\-1 & 1 & 3 & 5 \\2 & 3 & -1 & 0\end{bmatrix}##

a) Determine the range of L_A



2. Relevant equations

None



3. The attempt at a solution

The row-reduced matrix is as follows

##\begin{bmatrix}1 & 0 & -2 & -3 \\0 & 1 & 1 & 2 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}##



Then

##2x_{3}+3x_{4}##



##-x_{3}-2x_{4}##



##x_{3}##



##x_{4}##



Is this correct?







1. The problem statement, all variables and given/known data

##L \big( \begin{bmatrix}a & b \\c & d \end{bmatrix} \big) = \begin{bmatrix}0 & a \\b & c \end{bmatrix}##

a) Show that L is a linear transformation.

b) Define L^{k} = L \circ L^{k-1} for every integer k >= 2



2. Relevant equations

To be a linear transformation, these must be true

i) ##f(x_{1})+f(x_{2})=f(x_{1} + x_{2})##

ii) ##cf(x_{1})=f(cx_{1})##



3. The attempt at a solution

a) I'm not sure how to start showing this. For i) do I add the two matrices? For ii) do I just multiply each entry of the first matrix by c?

b) I don't know where to start at all. I'm not even sure what the question is asking.