5.2. THE MATRIX OF A LINEAR TRANSFORMATION 109

Proposition 5.2.9 Let A be an m×n matrix and consider it as a linear transformation bymultiplication on the left by A. Then the matrix M of this linear transformation with respectto the bases β = {u1, · · · ,un} for Fn and γ = {w1, · · · ,wm} for Fm is given by

M =(

w1 · · · wm

)−1A(

u1 · · · un

)where

(w1 · · · wm

)is the m×m matrix which has w j as its jth column. Note that

also (w1 · · · wm

)M(

u1 · · · un

)−1= A

Proof: Consider the following diagram.

AFn → Fm

qβ ↑ ◦ ↑ qγ

Fn → Fm

M

Here the coordinate maps are defined in the usual way. Thus

qβ (x)≡n

∑i=1

xiui =(

u1 · · · un

)x

Therefore, qβ can be considered the same as multiplication of a vector in Fn on the left

by the matrix(

u1 · · · un

). Similar considerations apply to qγ . Thus it is desired to

have the following for an arbitrary x ∈ Fn.

A(

u1 · · · un

)x=

(w1 · · · wn

)Mx

Therefore, the conclusion of the proposition follows. ■The second formula in the above is pretty useful. You might know the matrix M of a

linear transformation with respect to a funny basis and this formula gives the matrix of thelinear transformation in terms of the usual basis which is really what you want.

Definition 5.2.10 Let A ∈L (X ,Y ) where X and Y are finite dimensional vector spaces.Define rank(A) to equal the dimension of A(X) .

Lemma 5.2.11 Let M be an m×n matrix. Then M can be considered as a linear transfor-mation as follows.

M (x)≡Mx

That is, you multiply on the left by M.

Proof: This follows from the properties of matrix multiplication. In particular,

M (ax+by) = aMx+bMy ■

Note also that, as explained earlier, the image of this transformation is just the span of thecolumns, known as the column space.

The following theorem explains how the rank of A is related to the rank of the matrixof A.