Let V= Mat2×2(K) be a 4-dimensional vector space with the basis: v1=[1 0; 0 0], v2=[0 1; 0 0], v3=[0 0; 1 0], v4=[0 0; 0 1]. Let A=[a b; c d] ∈ Mat2×2(K). We define the linear map T:V→V, M→A·M, for M∈V.Find the matrix associated with T with respect to the standard basis.
My attempt: A*{v1 v2 v3 v4} to get a 2x8 matrix but i dont know where to go from here? i tried to put them together and got: [a 0 b 0; c 0 d 0; 0 a 0 b; 0 c 0 d] I have no idea if this is even correct?
Flatten the standard basis matrices $v_i$ to column vectors (just take the transpose of the row vectors you gave in the post), calculate each matrix product $A\cdot v_i$, and put the flattened result in the $i$th column.