Math 323
Change of Basis Notation
March 30, 2015
Since the notation in the book is not entirely the same (or as universally used) as what we
have done in class, I’ve summarized the important things below.
Suppose B = {v1 , . . . , vn } is a basis for a vector space V . We know that any vector v ∈ V
can be expressed uniquely in terms of B:
v = c1 v 1 + c2 v 2 + · · · + cn v n ,
where c1 , . . . , cn ∈ R are unique (i.e., once we choose v, the c’s are uniquely determined).
We will write
 
c1
 c2 
 
v = [v]B =  .. 
.
cn B
to mean that c1 , . . . , cn are the coordinates of v in terms of the basis B.
Transition Matrices on Rn : Using the standard basis S = {e1 , e2 , . . . , en } for Rn , then
our usual coordinates are the same as the coordinates with respect to S. So
   
x1
x1
 x2   x2 
   
 ..  =  ..  .
. .
xn S
xn
In this case if B = {v1 , . . . , vn } is another basis for Rn , then we let UB be the n × n matrix
formed by taking
.
..
.. 
..
.
.


UB = v1 v2 · · · vn  ,
..
..
..
.
.
.
where the columns of UB are obtained by taking the column vectors v1 , . . . , vn under the
standard basis and lining them up. The matrix UB is called the transition matrix or change
of basis matrix. Then for any vector v ∈ Rn we find formulas,
[v]S = UB · [v]B ,
and likewise
[v]B = UB−1 · [v]S ,
which allow us to change representations of v in terms of the standard basis and in terms of
the basis B and back again.
If A = {u1 , u2 , . . . , un } is another basis for Rn , then we can interchange between B-basis
representations and A-representations: for any v ∈ Rn ,
[v]A = UA−1 UB · [v]B
and likewise
[v]B = UB−1 UA · [v]A .
Matrices for Linear Transformations: Suppose that L : V → W is a linear transformation of finite dimensional vector spaces. If B = {v1 , v2 , . . . , vn } is a basis for V and
C = {w1 , w2 , . . . , wm } is a basis for W , then we can form the m × n matrix,
..
..
.
.

[L]CB = [L(v1 )]C [L(v2 )]C · · ·
..
..
.
.


..
.

[L(vn )]C  .
..
.
In this way, we can represent L by multiplication by the matrix [L]CB : for all vinV ,
L(v) C = [L]CB · [v]B .
To think about what this means: [L]CB takes as input vectors in coordinates with respect to
B, applies L to them, and then leaves as output vectors in coordinates with respect to C.
Example 1: Suppose that L : Rn → Rm is a linear transformation. Suppose that B =
{v1 , v2 , . . . , vn } is a basis for Rn and C = {w1 , w2 , . . . , wm } is a basis for Rm . Then taking
the m × n matrix [L]CB by In fact,
..
..
.
.

C
[L]B = [L(v1 )]C [L(v2 )]C · · ·
..
..
.
.


..
.

[L(vn )]C  ,
..
.
we have
[L(x)]C = [L]CB [x]B ,
∀x ∈ Rn .
Example 2: We know that any linear transformation L : Rn → Rm is represented by an
m × n matrix A:
L(x) = Ax,
∀x ∈ Rn ,
and
..
..
.
.

A = L(e1 ) L(e2 ) · · ·
..
..
.
.

.. 
.

L(en ) .
..
.
Everything here is done in terms of the standard basis. So what we have is
 .
..
.. 
..
.
.


L(e
)
L(e
)
·
·
·
L(e
=
A
=
[L]SSm

1
2
n ) .
n
..
..
..
.
.
.
Example 3: Let I : Rn → Rn be the identity transformation,
I(x) = x,
∀x ∈ Rn .
We can represent I in terms of a basis B = {v1 , v2 , . . . , vn }, and we find
.
..
.. 
..
.
.


[I]SB = UB = v1 v2 · · · vn  ,
..
..
..
.
.
.
where S is the standard basis on Rn . Likewise,
[I]BS = ([I]SB )−1 = UB−1 .
If L : Rn → Rn is a linear transformation, the representation of L in terms of the basis B is
[L]BB = [I]BS · [L]SS · [I]SB
= ([I]SB )−1 · [L]SS · [I]SB .
For any linear transformation, L : Rn → Rm , we can find [L]SSm
from Example 1 above. If B
n
n
m
is any basis of R and C is any basis of R ,
[L]CB = [I]CSm · [L]SSm
· [I]SBn .
n
Example 4: Let L : R3 → R3 be given by
 
   
1 2 3 x1
x1
L x2  =  0 1 0 x2  .
−1 0 9 x3
x3
nh −1 i h i h io
nh 0 i h i h io
2
4
0
1
0
1
Let B =
, 0 , 1 , and let C =
, 1 , 1 . Let S be the standard basis
−1
0
1
2
3
on R3 . Things you should check:


−1 2 4
[I]SB =  1 0 1 ;
0 1 2
and


1 2 3
[L]SS =  0 1 0 ;
−1 0 9
5


1
3
9
[I]CB =  2 −2 −3 .
−1 2
4


3
7 −3
[L]CS = −1 −1 −3 .
1
2
3
and
[L]CB = ??
[L]BC = ??