Harvey Mudd College Math Tutorial:
Change of Basis
Let V be a vector space and let S = {v1 , v2 , . . . , vn } be a set of vectors in V . Recall that S
forms a basis for V if the following two conditions hold:
1. S is linearly independent.
2. S spans V .
If S = {v1 , v2 , . . . , vn } is a basis for V , then every vector v ∈ V can be expressed uniquely
as a linear combination of v1 , v2 , . . . , vn :
v = c1 v 1 + c2 v 2 + · · · + cn v n .

Think of





c1
c2
..
.






as the coordinates of v relative to the basis S. If V has dimension n, then
cn
every set of n linearly independent vectors in V forms a basis for V . In every application,
we have a choice as to what basis we use. In this tutorial, we will desribe the transformation
of coordinates of vectors under a change of basis.
We will focus
vectors in R2 , although all of this generalizes to Rn . The standard basis
nh i on
h io
1
0
2
in R is 0 , 1 . We specify other bases with reference to this rectangular coordinate
system.
Let B = {u, w} and B 0 = {u0 , w0 } be two bases for R2 . For a vector v ∈ V , given its
coordinates [v]B in basis B we would like to be able to express v in tems of its coordinates
[v]B 0 in basis B 0 , and vice versa.
Suppose the basis vectors u0 and w0 for B 0 have the following coordinates relative to the
basis B:
This means that
"
a
b
#
"
c
d
#
0
[u ]B =
0
[w ]B =
u0 = au + bw
w0 = cu + dw
.
The change of coordinates matrix from B 0 to B
"
P =
a c
b d
#
governs the change of coordinates of v ∈ V under the change of basis from B 0 to B.
"
[v]B = P [v]B 0 =
a c
b d
#
[v]B 0 .
That is, if we know the coordinates of v relative to the basis B 0 , multiplying this vector by
the change of coordinates matrix gives us the coordinates of v relative to the basis B.
Why?
The transition matrix P is invertible. In fact, if P is the change of coordinates matrix from
B 0 to B, the P −1 is the change of coordinates matrix from B to B 0 :
[v]B 0 = P −1 [v]B
Example
nh i h io
nh i h
io
Let B = 10 , 01 and B 0 = 31 , −2
.
1
0
The change of basis matrix form B to B
is
"
#
3 −2
P =
.
1 1
h i
The vector v with coordinates [v]B 0 =
relative to the basis B 0 has coordinates
"
[v]B =
3 −2
1 1
#"
2
1
#
"
=
4
3
2
1
#
relative to the basis B. Since
1
5
"
P
−1
=
− 51
2
5
3
5
#
,
we can verify that
"
[v]B 0 =
1
5
− 15
2
5
3
5
#"
4
3
#
"
=
2
1
#
which is what we started with.
In the following example, we introduce a third basis to look at the relationship between two
non-standard bases.
Example
nh i h io
Let B 00 = 21 , 14 . To find the change
of coordinates matrix from the basis B 0 of
00
the previous example to
h iB , we
h first
i ex3
press the basis vectors 1 and −2
of B 0
1
as
of the basis vectors
h ilinearhcombinations
i
2
1
00
and
of
B
:
1
4
"
Set
"
3
1
#
−2
1
#
"
= a
"
= c
2
1
#
2
1
#
"
1
4
#
"
1
4
#
+b
+d
and solve the resulting systems
of r a, b, c, and d:
"
#
"
#
"
#
11 2
1 1
=
−
7 1
7 4
"
#
"
#
"
#
−9 2
4 1
−2
=
+
1
1
7
7 4
3
1
Thus, the transition matrix form B 0 to B 00 is
−9
7
4
7
" 11
7
−1
9
The vector v with coordinates
h i
2
1
#
.
relative to the basis B 0 has coordinates
" 11
7
−1
9
−9
7
4
7
#"
2
1
" 13 #
#
7
2
7
=
relative to the basis B 00 . This is, back in the standard basis,
13
[v]B =
7
"
2
1
#
2
+
7
"
1
4
#
which agrees with the results of the previous example.
"
=
4
3
#
,
Rotation of the Coordinate Axes
Suppose we obtain a new coordinate system from the standard rectangular coordinate system by rotating the axes counterclockwise by an angle θ. The new basis
B 0 = {u0 , v0 } of unit vectors along the x0 and y 0 -axes, respectively, has coordinates
"
cos θ
sin θ
"
− sin θ
cos θ
0
[u ]B =
[v0 ]B =
#
#
in the original coordinate system.
"
#
"
#
h i
cos θ − sin θ
cos θ sin θ
Thus, P =
and P −1 =
. A vector xy
in the original
B
sin θ cos θ
− sin θ cos θ
h 0i
coordinate system has coordinates xy0 0 given by
B
"
x0
y0
#
"
=
B0
cos θ sin θ
− sin θ cos θ
in the rotated coordinate system.
Example
h i
The vector [v]B = 32 in the original coordinate system has coordinates

[v]B 0 = 
√
2
2√
−
2
2
√
2
√2
2
2
"

3
2
#

=
√
5 2
2√
− 22


in the coordinate system formed by rotating the axes by 45◦ .
In the following Exploration, set up your
own basis in R2 and compare the coordinates of vectors in your basis to their coordinates in the standard basis.
Exploration
#"
x
y
#
B
Key Concepts
h i
h i
Let B = {u, v} and B 0 = {u0 , v0 } be two bases for R2 . If [u]B = ab and [v]B = dc , then
"
#
a c
P =
is the change of coordinates matrix from B 0 to B and P −1 is the change
b d
of coordinates matrix from B to B 0 . That is, for any v ∈ V ,
[v]B = P [v]B 0
[v]B 0 = P −1 [v]B .
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