Set M to be the set of all matrices of the form: $\bigl( \begin{smallmatrix} 0 & b \\ c & 0 \end{smallmatrix} \bigr)$, with b, c being real numbers.
The basis for M (given) is $\epsilon$ = $\{[\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}] ,[\begin{smallmatrix} 0 & 0 \\ 1 & 0 \end{smallmatrix}] \}$, with the elements referred to as $\epsilon = \{e_2, e_3\}$
Find the coordinate vector of A = $[\begin{smallmatrix} 0 & 2 \\ 70 & 0 \end{smallmatrix}]$ in the basis $\epsilon$.
I'm not really sure how to approach this problem, as I've never seen matrices being used as parts of a basis. I read on this question that I can think of these as 4-tuples, so there's nothing "special" going on, so is the answer simply:
$[A]_\epsilon$ = $[\begin{smallmatrix} 2 \\ 70 \end{smallmatrix}]$ since $A = 2e_2 + 70e_3$?
Best Answer
Yes that is true. The coordinates are just the coefficients of the base vectors in the linear combination $2e_1+70e_2$ that results in your given vector $A=2e_1+70e_2$.