Here is the problem:

Given the linear transformation T: R3 -> R3, such that T(V)=kV, where k is a constant. Find the matrix(A) of T reltve to th bases B and B' for the following:

a) B=standard basis of R3=B'

b) B=standard basis of R3, B'={(1,0,0),(0,1,1),(0,1,1)}

c) Verify [T(1,2,3)]B' = A[(1,2,3)]B

Ok, here is what I did so far:

For part a): A=[ 1 0 0

0 1 0

0 0 1] this is the standard basis for basis of R3

and therefore I get T(v)=[k 0 0

0 k 0

0 0 k]. It seems quite weird to me. But I could not tell where is wrong.

For part b): since the standard basis for B is{(1,0,0) (0,1,0) (0,0,1)}, and B'=={(1,0,0),(0,1,1),(0,1,1)}

T(1,0,0) = (k,0,0)

T(0,1,0)= (0,k,0)

T(0,0,1)=(0,0,k) for each of these three, write them in terms of B', however, unlike the usual problem dealing with numbers, so I am stuck on this. What to do next?