8.3. THE MATRIX OF A LINEAR TRANSFORMATION 221

Then it follows that B is the matrix of L with respect to the basis

{qγe1, · · · , qγen} ≡ {w1, · · · , wn}.

That is, A and B are matrices of the same linear transformation L. Conversely, sup-pose whenever V is an n dimensional vector space, there exists L ∈ L (V, V ) and bases{v1, · · · , vn} and {w1, · · · , wn} such that A is the matrix of L with respect to {v1, · · · , vn}and B is the matrix of L with respect to {w1, · · · , wn}. Then it was shown above thatA ∼ B. ■

What if the linear transformation consists of multiplication by a matrix A and you wantto find the matrix of this linear transformation with respect to another basis? Is there aneasy way to do it? The next proposition considers this.

Proposition 8.3.10 Let A be an m×n matrix and let L be the linear transformation whichis defined by

L

(n∑

k=1

xkek

)≡

n∑k=1

(Aek)xk ≡m∑i=1

n∑k=1

Aikxkei

In simple language, to find Lx, you multiply on the left of x by A. (A is the matrix of Lwith respect to the standard basis.) Then the matrix M of this linear transformation withrespect to the bases β = {u1, · · · ,un} for Fn and γ = {w1, · · · ,wm} for Fm is given by

M =(

w1 · · · wm

)−1

A(

u1 · · · un

)where

(w1 · · · wm

)is the m×m matrix which has wj as its jth column.

Proof: Consider the following diagram.

L

Fn → Fm

qβ ↑ ◦ ↑ qγFn → Fm

M

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

(x1 · · · xn

)T≡

n∑i=1

xiui.

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. ■In the special case where m = n and F = C or R and {u1, · · · ,un} is an orthonormal

basis and you wantM , the matrix of L with respect to this new orthonormal basis, it followsfrom the above that

M =(

u1 · · · um

)∗A(

u1 · · · un

)= U∗AU

where U is a unitary matrix. Thus matrices with respect to two orthonormal bases areunitarily similar.

8.3. THE MATRIX OF A LINEAR TRANSFORMATION 221Then it follows that B is the matrix of L with respect to the basis{qyei,-°° + FyEn } = {wi,-: ,Wn}.That is, A and B are matrices of the same linear transformation DL. Conversely, sup-pose whenever V is an n dimensional vector space, there exists L € £(V,V) and bases{v1,-++ Un} and {w1,--- ,wn} such that A is the matrix of L with respect to {v1,--- , Un}and B is the matrix of L with respect to {w1,--- ,wWn,}. Then it was shown above thatA~B.tWhat if the linear transformation consists of multiplication by a matrix A and you wantto find the matrix of this linear transformation with respect to another basis? Is there aneasy way to do it? The next proposition considers this.Proposition 8.3.10 Let A be anmxn matrix and let L be the linear transformation whichis defined byL (>: ve] — S- (Aex) oo S- SS Ajpepe;k=1 k=1 i=1 k=1In simple language, to find Lx, you multiply on the left of x by A. (A is the matrix of Lwith respect to the standard basis.) Then the matrix M of this linear transformation withrespect to the bases B = {u,,--: ,u,} for F” and y = {wi,--: ,Wm} for F™ is given by-1w=(w we) AC ue)where ( Wi -'+) Wm ) is them x m matriz which has w; as its gj column.Proof: Consider the following diagram.LFr" + -F”qt °o THFr” —+ -F”MHere the coordinate maps are defined in the usual way. ThusT ni=1Therefore, gg can be considered the same as multiplication of a vector in F” on the left bythe matrix ( Uc: Up, ) . Similar considerations apply to q,. Thus it is desired to havethe following for an arbitrary x € F”.A( wy vee un )x= (wi vee wy, ) MxTherefore, the conclusion of the proposition follows.In the special case where m = n and F = C or R and {w,--- ,u,} is an orthonormalbasis and you want M, the matrix of L with respect to this new orthonormal basis, it followsfrom the above thatM=(w “Um )A(m vee uy) =U*AUwhere U is a unitary matrix. Thus matrices with respect to two orthonormal bases areunitarily similar.