Description
In class, we continued to discuss how we to use matrix representations of linear maps between
vector spaces. Suppose we define the horrible map you came up with in class, T : R
4 → R
3
• T(1, 2, 3, 4) = (5, 6, 7)
• T(11, 10, 9, 8) = (2, 3, 1)
• T(1, 5, 7, 2) = (7, 8, 6)
• T(0, 0, 0, 1) = (9, 1, 1)
Our strategy for handling this is to define a map U : R
4 → R
4
that translates from a nice basis
{(1, 0, 0, 0),(0, 1, 0, 0), …} to the basis upon which T is defined: {(1, 2, 3, 4),(11, 10, 9, 8), …}.
We did this in class and defined the matrix representation for U to be
M(U) =
25
24
−17
6
15
8
0
1
24
1
6
−1
8
0
−1
2
1
−1
2
0
−7
2
8
−11
2
1
If we define the set of mapped elements in R
3
to be {(5, 6, 7),(2, 3, 1),(7, 8, 6),(9, 1, 1)}, then
the matrix representation of T is
M(T) =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
But this is not enough to get the full answer. Why? Suppose there is a vector v for which
we apply M(U) and then M(T) and we get the result (1, 0, 1, 1). This vector represents:
1 · (5, 6, 7) + 0 · (2, 3, 1) + 1 · (7, 8, 6) + 1 · (9, 1, 1). Ideally we’d like the result to actually be
our answer directly and not have to do this extra step. So we can define a map S that goes
from {(5, 6, 7),(2, 3, 1),(7, 8, 6),(9, 1, 1)} to {(1, 0, 0),(0, 1, 0),(0, 0, 1)}.
Define M(S).