Question

if M (x,y) = (x+y, x+y ) 

Find the matrix representation of M with respect to (v1,v2) where v1= (1,1) v2 = (1,3) 

Answer 1

Consider a vector $l=xv_1+yv_2=(x+y,x+3y)$. We act on a vector with a matrix $M$ and receive: $(2x+4y,2x+4y)=(2x+4y)v_1$. Thus, the matrix $M$ with respect to $(v_1,v_2)$ has a representation: $M_{v_1,v_2}=\left(\begin{array}{cc}2&4\\0&0\end{array}\right)$.