Originally Posted by
HallsofIvy But to generalize: If L is a linear transformation from vector space U, with dimension n, to vector space V, with dimension m, then it can be written in matrix form for given basis
of U and given basis
of V (changing bases will result in different but "equivalent" matrices representing the same transformation).
Take
and write it as a linear combination of the basis
, which you can do since
is in V. Say,
. Then
is the first column in the matrix. Similarly, writing
as a linear combination of
gives the second column, etc.
I'm new to a lot of this stuff so some of that definition didn't really make sense to me... You are saying that i would have to approach a problem like the one below differently, correct?
(b) Let
L : R3 ->R2 be the linear transformation defined by
L
(x, y, z) = (x + y + z, y + z)
(i) Find the standard (2 × 3) matrix representation of L and compute L(0,−1, 1).