- #1
says
- 594
- 12
Homework Statement
Given the linear transformations
f : R 3 → R 2 , f(x, y, z) = (2x − y, 2y + z), g : R 2 → R 3 , g(u, v) = (u, u + v, u − v), find the matrix associated to f◦g and g◦f with respect to the standard basis. Find rank(f ◦g) and rank(g ◦ f), is one of the two compositions an isomorphism? If yes find its inverse.
Homework Equations
f=
[2 -1 0]
[0 2 1]
g=
[1 0]
[1 1]
[1 -1]
The Attempt at a Solution
f◦g =
[1 -1]
[3 1]
g◦f=
[ 2 -1 0 ]
[ 2 1 1 ]
[ 2 -3 -1 ]I row reduced both matrices, I don't want to write them out here, but the rank (g ◦ f) = 2, it has one free variable. Rank(f ◦g) = 2. I think the real question I have here is with the last part of the question.
'Is one of the two compositions an isomorphism?' For a linear transformation to be an isomorphism is has to be injective and surjective. Is the very nature that this L.T maps from R3 to R2 and vice versa reason enough to say it is not an isomorphism?
I found an inverse of
f◦g =
[1/4 1/4]
[-3/4 1/4]