Changing Bases

Goals: Understand the concept of coordinates. Know how to find and use a transition matrix to find the coordinates of a vector when changing from one basis to another.

Since the choice of a basis for a vector space is not unique, we are faced with the set of problems associated with choosing a basis. In order to solve these types of problems we need to refine what we mean by the coordinates of a vector.

I. Coordinate Vectors

From what we proved with the uniqueness theorem, a basis B of a vector space V gives us a unique way to represent any vector in V. We can think of B as a "coordinate system" for V.

To see this, we begin with an example in R².

Let

and

be vectors in R² with

Do u and v form a basis for R²?

Since u and v are a basis for R², we might think of the scalar multiples of u and v as a kind of a grid-work or "coordinate system" for R². That is every point in R² may be written as a linear combination of u and v.

What are the likely coordinates of a point P in this system?

Definition Coordinates: Let V be a finite-dimensional vector space with an ordered basis
B = {v₁, v₂, v₃, ... , v_n}, then for each vector v ÎV , there exist unique scalars c₁, c₂, .... , c_n such that

v = c₁v₁ + c₂v₂ + ... +c_nv_n

The vector c = (c₁, c₂, .... , c_n )^T is called the coordinate vector of v with respect to B and is denoted [v]_B. The c_i's are called the coordinates of v relative to B.

An ordered basis refers to the ordering of the basis vectors that is kept constant when writing coordinates. For example, when we think of a point in R², we always refer to it's (x,y) coordinates and not (y,x) coordinates. This is the standard ordering of coordinates in the plane.

II. Changing Coordinates in R²

The standard basis E for R² is {e₁, e₂}. Any vector v in R² can be written as a linear combination

where x and y are scalars. The point P represented on the plane by the vector v has coordinates (x, y). Not much new here.
Now suppose we choose another basis for R², say B = {v₁, v₂}. Then

for some scalars a and b. The coordinate vector [v]_B = (a, b). Not too surprising, different bases generate different coordinate vectors.

What are the coordinates of P in the standard basis?

What are the coordinates of P using the ordered basis {u , v}?

Graphically, we may simply read the coordinates, P(8,10) or P(2,2)_B

We will see before long that changing coordinates from one basis to another basis is a very useful operation in solving certain types of problems. For example, systems of differential equations are sometimes solvable under a change of coordinates. The technique is important enough for further study. We begin with:

Consider the following two Change of Coordinates problems in R² with the ordered bases
B = {u₁, u₂} and E = {e₁, e₂}.

I. Given the vector [v]_B = (c₁, c₂)^T find the vector [v]_E =(x₁, x₂)^T.
In this case v = c₁u₁ +c₂u₂ and we want to find v = x₁e₁ + x₂e₂.

II. Given a coordinate vector [v]_E find the vector [v]_B.
We are given the coordinates of a vector in the standard basis E, what are it's coordinates with respect to B = {u₁, u₂}, that is we have [v]_E =(x₁, x₂)^T and we want [v]_B = (c₁, c₂)^T.

Find [v]_E

given

with respect to the basis B = { u₁ , u₂},

B

Thus,

And

,

(These are in the standard basis E)

Solution: In this case we are given [v]_B = (c₁, c₂), the coordinates of v in the ordered basis {u₁, u₂}.

Further we have,

or equivalently,

and

or

making the substitution for u₁ and u₂,

This simplifies to:

The corresponding coordinate vector [v]_E is

[v]_E =

The coordinate vectors satisfy the matrix equation.

[v]_E =

[v]_B

Look at the matrix carefully. What are the column vectors?

The column vectors are u₁ and u₂. Is this just a coincidence?

Let

then given any coordinate vector [v]_B, the corresponding coordinate vector [v]_E is [v]_E = U [v]_B .

This completely solves Problem I.

Given the following coordinate vectors

with the ordered basis

and

find [v]_E .

(i.e. given , find
)

For

try changing these coordinates

We can write this last equation as

where x = [v]_E and C = [v]_B

The matrix U is called the transition matrix from the ordered basis B to the ordered basis E.

Now consider the equation .

What do you suppose that this equation says?

The second problem was:

II. Given a coordinate vector [v]_E find the vector [v]_B.
Here we switch from the standard basis E to B = {u₁, u₂}, that is we have [v]_E =(x₁, x₂)^T and we want [v]_B = (c₁, c₂).

This problem is completely solved by the matrix equation .
In this case U^-1 is the transition matrix from [v]_E to [v]_B.

Let

and

and

Find the coordinates of x with respect to u₁ and u₂.

Solution: we are given

we need to find c₁ and c₂ such that:

From above we have

that is [v]_B = (3, -2)^T

or equivalently

As in the next problem, sometimes the vectors are not called a basis, in which case you may need to check.

Let

and

Find the transition matrix from {e₁, e₂} to {b₁, b₂} and determine the coordinates of with respect to {b₁, b₂}.

U is the transition matrix from {b₁, b₂} to {e₁, e₂}. Hence the transition matrix from {e₁, e₂} to {b₁, b₂} is

and

or

that is

III. The Transition Matrix from {v₁, v₂} to {u₁, u₂}

Our plan is to extend these ideas into any vector space V. So far, the transition matrix U as we have developed it, only works from a basis {b₁, b₂} to the standard basis {e₁, e₂} in R².

How do we find a transition matrix for any two bases in R²?

Given V = {v₁, v₂} and U = {u₁, u₂} be two bases for R².

Find the transition matrix corresponding to the change of basis
from V to U.

From the previous section, we saw how to find the transition matrix V from V to E. Also, we saw how to find the transition matrix U^-1 from E to U.

How can we use this to find a transition matrix from V to U? Here is one solution. First, we change coordinates from V to E, and then from E to U.

This graphic illustrates the path from V to U through E.

Thus, the matrix product U^-1V is the transition matrix from V to U.

Find the transition matrix M corresponding to the change of basis from {v₁, v₂} to {u₁, u₂}, where

and

Solution:

Find

and

we compute

Now we take these ideas a little bit further and look at the general case.

IV. The Change of Basis for a Vector Space V

The previous example easily generalizes to any n-dimensional vector Space V.

Theorem: Let V be an n dimensional vector space with ordered bases B = { b₁, ... , b_n} and U = {u₁, ... , u_n} . Then there exists an nxn transition matrix S such that

[v]_U = S [v]_B

Furthermore, if V = (b₁, b₂, ... , b_n) and U = (u₁, u₂, ... , u_n) are two matrices formed by the column vectors of B and U,
then S = U^-1V

Proof:

Time for some practice on our four vector spaces.

Find the transition matrix representing the change of coordinates on P₃ from the ordered basis E = [1, x, x²] to the ordered basis B = [1, 1 + x, 1 + x + x²].

Solution:

(Hint: Write the vectors in B with respect to E.)

or

or

or

To be continued...

Let E = [v₁, v₂, v₃] = [(1, 1, 1)^T, (2, 3, 2)^T, (1, 5, 4)^T]
and F = [u₁, u₂, u₃] = [(1, 1, 0)^T, (1, 2, 0)^T, (1, 2, 1)^T]

a) Find the transition matrix from E to F.

b) If x = 3v₁ + 2v₂ -v₃ find [x]_F

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Lecture 9 Notes