Recall that we can answer questions about vectors in an abstract vector space (like whether they’re linearly independent) by finding a basis for the space, moving the vectors into Rn using the basis, answering the question there, and then moving back into the abstract space.
Can we do the same for linear transformations?
Let V,W be vector spaces.
Let B={b1,...,bn} be an ordered basis for V and C={c1,...,cm} be an ordered basis for W.
Let T:V→W be a linear transformation.
We want to find some m×n matrix M such that [T(x)]C=M[x]B. If we do this, then we can completely represent T(x) as a transformation from Rn to Rm rather than as an abstract transformation.
Assume that x=r1b1+...+rnbn.
[x]BT(x)[T(x)]C=⎝⎜⎛r1...rn⎠⎟⎞=T(r1b1+...+rnbn)=r1T(b1)+...+rnT(bn)=r1[T(b1)]C+...+rn[T(bn)]C=[[T(b1)]C...[T(bn)]C]⎝⎜⎛r1...rn⎠⎟⎞=[[T(b1)]C...[T(bn)]C][x]B
M=[[T(b1)]C...[T(bn)]C].
We call M the matrix for T relative to the bases B and C.
Ex. Suppose that B={b1,b2} is an ordered basis for V, C={c1,c2,c3} is an ordered basis for W, and T:V→W is a linear transformation.
T(b1)T(b2)=3c1+2c2+4c3=c1+5c2+3c3 Find a matrix M for T relative to B and C.
[T(b1)]C[T(b2)]CM=⎝⎜⎛324⎠⎟⎞=⎝⎜⎛153⎠⎟⎞=⎝⎜⎛324153⎠⎟⎞
You can see that once the bases B and C are fixed, we can find a unique matrix for any linear transformation between V and W.
Now we’re going to look at a special case: V=W, B=C.
In this case, we call M the matrix for T relative to B, or simply the B-matrix for T, and denote it [T]B (i.e. [T(x)]B=[T]B[x]B).
Ex. We can represent taking the derivative as a linear transformation.
T:P3→P3T(a0+a1t+a2t2+a3t3)=0a0+a1+2a2t+3a3t2
Find the B-matrix for T where B={1,t,t2,t3}.
[T(1)]B[T(t)]B[T(t2)]B[T(t3)]B=⎝⎜⎜⎜⎛0000⎠⎟⎟⎟⎞=⎝⎜⎜⎜⎛1000⎠⎟⎟⎟⎞=⎝⎜⎜⎜⎛0200⎠⎟⎟⎟⎞=⎝⎜⎜⎜⎛0030⎠⎟⎟⎟⎞[T]B=[[T(1)]B[T(t)]B[T(t2)]B[T(t3)]B]=⎝⎜⎜⎜⎛0000100002000030⎠⎟⎟⎟⎞
Theorem. Suppose A=PCP−1. If B is a basis for Rn formed from the columns of P, then C is the B-matrix for x→Ax.
Recall that Rn has infinitely many bases – we just tend to use the standard basis because it’s computationally convenient. However, you may encounter data represented in a different basis. We can use C in that case to change coordinates between different basis systems.