change of basis - hmc calculus tutorial

8/9/2019 Change of Basis - HMC Calculus Tutorial

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Change of Basis

Let V be a vector space and let S v be a set of vectors in V . Recall that S forms a

basis for V if the following two conditions hold:

S is linearly independent.1.

S spans V .2.

If S v is a basis for V , then every vector v can be expressed uniquely as a

linear combination of v :

v v v v

Think of as the coordinates of v relative to the basis S . If V has dimension n, then 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 on vectors in R , although all of this generalizes to R . The standard basis in R is

. We specify other bases with reference to this rectangular coordinate system.

Let B u and B u be two bases for R . For a vector v , given its coordinates

[v] in basis B we would like to be able to express v in tems of its coordinates [v] in basis B , and

vice versa.

Suppose the basis vectors u and w for B have the following coordinates relative to the basis B:

This means that

= 1 v2 vn

= 1 v2 vn V

1 v2 vn

= c1 1 + c2 2 + + cn n

c1

c2

cn

2 n 2

01

10

= w = w 2 V

B B

[u ] B

[w ] B

=

=

a

b

c

d

u

w

=

=

au w+ b

cu w+ d

nge of Basis - HMC Calculus Tutorial file:///D:/2015.1/Emilia/Change of Basis - HMC Calculus Tutorial.html

5 05/05/2015 10:34



The change of coordinates matrix from B to B

governs the change of coordinates of v under the change of basis from B to B.

That is, if we know the coordinates of v relative to the basis B , 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 to B , the

P is the change of coordinates matrix from B to B :

[v] [v]

Example

Let and . The

change of basis matrix form B to B is

The vector v with coordinates relative to

the basis B has coordinates

relative to the basis B . Since

we can verify that

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.

P = a

b

c

d

V

[v] [v] v] B = P B = a

b

c

d [ B

−1

B = P −1 B

B =01

10 B =

13

1−2

P = 31

−

21

[v] B =12

[v] B =3

1

−2

1

2

1=

4

3

P −1 = 51

−51

52

53

[v] B = 51

−51

52

53

4

3=

2

1


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Example

Let . To find the change of

coordinates matrix from the basis B of the previous

example to B , we first express the basis vectors

and of B as linear combinations of the basis

vectors and of B :

and solve the resulting systems of r a and d :

Thus, the transition matrix form B to B is

The vector v with coordinates relative to the basis B has coordinates

relative to the basis B . This is, back in the standard basis,

which agrees with the results of the previous example.

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 . The new basis

B of unit vectors along the x - and y -axes,

respectively, has coordinates

B =12

41

13

1

−2

12

41

Set3

1

−2

1

=

=

a2

1+ b

1

4

c2

1+ d

1

4

b c

3

1

−2

1

=

=

7

11 2

1−

7

1 1

4

7

−9 2

1+

7

4 1

4

711

9−

1

7−9

74

12

711

9−1

7−9

74

2

1= 7

13

72

[v] B =7

13 21

+72 1

4= 4

3

= u v


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in the original coordinate system.

Thus, and

. A vector in the original coordinate system has coordinates

given by

in the rotated coordinate system.

Example

The vector in the original coordinate system

has coordinates

in the coordinate system formed by rotating the axes by

45 .

In the following Exploration, set up your own basis in Rand compare the coordinates of vectors in your basis to

their coordinates in the standard basis.

Exploration

Key Concepts

Let B u and B u be two bases for R . If and , then

is the change of coordinates matrix from B to B and P is the change of

coordinates matrix from B to B . That is, for any v ,

[u ] B

[v ] B

=

=

cos

sin

− sin

cos

P =cos

sin

− sin

cos

P −1 =cos

− sin

sin

cos y x

B

x y B

x

y B

=cos

− sin

sin

cos

x

y B

[v] B =23

[v] B =

22

− 22

22

22

3

2 =

25 2

− 22

2

= v = v 2 [u] B =ba [v] B = c

d

P = a

b

c

d −1

V


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[I'm ready to take the quiz.] [I need to review more.]

[Take me back to the Tutorial Page.]

[v] B

[v] B

=

=

P [v] B

P [v]−1 B


change of basis - hmc calculus tutorial

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