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Similarity
We start with a linear transformation L : V → W and with bases
X = x1 , . . . , xn of V and Y = y1 , . . . , ym of W . Recall that every
vector v ∈ V can be written in exactly one way as a linear combination of the basis vectors of X:
v = c1 x1 + · · · + cn xn ,
where all ci ∈ R; we call the n × 1 column vector (c1 , . . . , cn )> the
coordinate list of v wrt X (with respect to X). Of course, every
vector w ∈ W has a coordinate list wrt Y .
Linear transformations defined on Rn are easy to describe.
Theorem 0.1
If L : Rn → Rm is a linear transformation, then there exists a unique
m × n matrix A such that
L(y) = Ay
for all y ∈ Rn (here, y is an n × 1 column matrix and Ay is matrix
multiplication).
Proof. If e1 , . . . , en is the standard basis of Rn and e01 , . . . , e0m is
the standard basis of Rm , define A = [aij ] to be the matrix whose
jth column is the coordinate list of L(ej ). If S : Rn → Rm is
defined by
P S(y) = Ay, then S = L because both agree on a basis:
L(ej ) = i aij ei = Aej . Uniqueness of A follows from the fact that
if L(y) = By for all y, then Bej = L(ej ) = Aej for all j; that is, the
columns of A and B are the same.
Theorem 0.1 establishes the connection between linear transformations and matrices, and the definition of matrix multiplication
arises from applying this construction to the composite of two linear
transformations.
Definition
Let X = v1 , . . . , vn be a basis of V and let Y = w1 , . . . , wm be
a basis of W . If L : V → W is a linear transformation, then the
matrix of L is the m × n matrix A = [aij ] whose jth column
a1j , a2j , . . . , amj is the coordinate list of L(vj ) wrt Y : L(vj ) =
P
m
i=1 aij wi = a1j w1 + a2j w2 + · · · + amj wm .
Since the matrix A depends on the choice of bases X and Y , we
will write
A = Y [L]X
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when it is necessary to display them. Varying any of them most likely
changes the matrix.
Consider the important special case LA : Rn → Rm given by
LA (y) = Ay, where A is an m × n matrix and y is an n × 1 column
vector. If E = e1 , . . . , en and E 0 = e01 , . . . , e0m are the standard
bases of Rn and Rm , respectively, then the definition of matrix
multiplication says that LA (ej ) = Aej is the jth column of A. But
Aej = a1j e01 + a2j e02 + · · · + amj e0m ;
that is, the coordinates of LA (ej ) = Aej with respect to the basis
e01 , . . . , e0m are (a1j , . . . , amj ). Therefore, the matrix associated to LA
is the original matrix A:
E 0 [LA ]E
= A.
In case V = W , we often let the bases X = v1 , . . . , vn and
Y = w1 , . . . , wm coincide. If 1V : V → V , given by v 7→ v, is the
identity linear transformation, then X [1V ]X is the n × n identity
matrix I, which has 1’s on the diagonal and 0’s elsewhere. On the
other hand, if X and Y are different bases, then Y [1V ]X is not the
identity matrix. The matrix Y [1V ]X is called the transition matrix
from X to Y ; its columns are the coordinate lists of the v’s with
respect to the w’s.
Example
1. Let X = e1 , e2 = (1, 0), (0, 1) be the standard basis of R2 . If L : R2 → R2
is rotation by 90◦ , then L : e1 7→ e2 and e2 7→ −e1 . Hence, the matrix of L
relative to X is
"
#
0 −1
.
X [L]X =
1 0
If we re-order X to obtain the new basis Y = η1 , η2 , where η1 = e2 and
η2 = e1 , then L(η1 ) = L(e2 ) = −e1 = −η2 and L(η2 ) = L(e1 ) = e2 = η1 .
The matrix of L relative to Y is
"
#
0 1
.
Y [L]Y =
−1 0
2. Let L : R2 → R2 be rotation by α degrees about the origin. Take X = Y
to be the standard basis e1 = (1, 0), e2 = (0, 1). Now L(e1 ) = (cos α, sin α)
and L(e2 ) = (cos(90 + α), sin(90 + α) = (− sin α, cos α). Thus the matrix of
L wrt standard basis is
"
#
cos α − sin α
.
X [L]X =
sin α cos α
2
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3. If L : C → C is complex conjugation, and if we choose the bases X = Y =
1, i, then the matrix is
"
#
1 0
.
X [L]X =
0 −1
The next theorem shows where the definition of matrix multiplication comes from: the product of two matrices is the matrix of a
composite.
Theorem 0.2
Let L : V → W and S : W → U be linear transformations. Choose
bases X = x1 , . . . , xn of V , Y = y1 , . . . , ym of W , and Z = z1 , . . . , z`
of U . Then
Z [S ◦ L]X = Z [S]Y
Y [L]X ,
where the product on the right is matrix multiplication.
Proof. Let Y [L]X = [aij ], so
P that L(xj ) =
Z [S]Y = [bqp ], so that S(yp ) =
q bqp zq . Then
X
S ◦ L(xj ) = S(L(xj )) = S
apj yp
P
p apj yp ,
and let
p
=
X
apj S(yp ) =
XX
p
where cqj =
P
p bqp apj .
Z [S
p
apj bqp zq =
q
X
cqj zq ,
q
Therefore,
◦ L]X = [cqj ] =
Z [S]Y
Y [L]X
.
Corollary 0.3
Matrix multiplication is associative.
Proof. Let A be an m × n matrix, let B be an n × p matrix, and
let C be a p × q matrix. There are linear transformations
L
L
L
C
B
A
Rq →
Rp →
Rn →
Rm
We know C = [LC ], B = [LB ], and A = [LA ].
Then
[LA ◦ (LB ◦ L)] = [LA ][LB ◦ LC ] = [LA ]([LB ][LC ]) = A(BC).
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On the other hand,
[(LA ◦ LB ) ◦ LC ] = [LA ◦ LB ][LC ] = ([LA ][LB ])[LC ] = (AB)C.
Since composition of functions is associative, LA ◦ (LB ◦ LC ) =
(LA ◦ LB ) ◦ LC , and so
A(BC) = [LA ◦ (LB ◦ LC )] = [(LA ◦ LB ) ◦ LC ] = (AB)C.
The connection with composition of linear transformations is the
real reason why matrix multiplication is associative.
Corollary 0.4
Let L : V → W be a linear transformation of vector spaces V and W ,
and let X and Y be bases of V and W , respectively. If there is a linear
transformation T : W → V with L ◦ T = 1V , then the matrix of T is
the inverse of the matrix of L :
X [T ]Y
= (Y [L]X )−1 .
Proof. We have I = Y [1W ]Y =
Y [L]X .
X [1V ]X = X [T ]Y
Y [L]X
X [T ]Y
, and so I =
The next corollary determines all the matrices arising from the
same linear transformation as we vary bases. Here we take the same
basis fore and aft: X = Y .
Corollary 0.5
Let L : V → V be a linear transformation on a vector space V . If X
and Y are bases of V , then there is a nonsingular matrix P (namely,
the transition matrix P = Y [1V ]X ) with entries in R so that
−1
.
Y [L]Y = P X [L]X P
Conversely, if B = P AP −1 , where B, A, and P are n × n matrices
with P nonsingular, then there is a linear transformation L : Rn →
Rn and bases X and Y of Rn such that B = Y [L]Y and A = X [L]X .
Proof. The first statement follows from Theorem ?? and associativity:
Y [L]Y
= Y [1V L1V ]Y = (Y [1V ]X )(X [L]X )(X [1V ]Y ).
Set P = Y [1V ]X and note that Corollary ?? gives P −1 = X [1V ]Y .
4
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The converse is a bit longer.
Definition
Two n × n matrices B and A with entries in R are similar if there
is a nonsingular matrix P with entries in R such that B = P AP −1 .
Thus, two matrices arise from the same linear transformation on
a vector space V (from different choices of bases) if and only if they
are similar.
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