# 2.5 The Change of Coordinate Matrix

In many areas of mathematics, a change of variable is used to simplify the appearance of an expression. For example, in calculus an antiderivative of  can be found by making the change of variable . The resulting expression is of such a simple form that an antiderivative is easily recognized:



Similarly, in geometry the change of variable



can be used to transform the equation  into the simpler equation , in which form it is easily seen to be the equation of an ellipse. (See Figure 2.4.) We will see how this change of variable is determined in Section 6.5. Geometrically, the change of variable



is a change in the way that the position of a point P in the plane is described. This is done by introducing a new frame of reference, an -coordinate system with coordinate axes rotated from the original -coordinate axes. In this case, the new coordinate axes are chosen to lie in the direction of the axes of the ellipse. The unit vectors along the -axis and the -axis form an ordered basis



for , and the change of variable is actually a change from , the coordinate vector of P relative to the standard ordered basis , to , the coordinate vector of P relative to the new rotated basis .

A natural question arises: How can a coordinate vector relative to one basis be changed into a coordinate vector relative to the other? Notice that the system of equations relating the new and old coordinates can be represented by the matrix equation



Notice also that the matrix



equals , where I denotes the identity transformation on . Thus  for all . A similar result is true in general.

# Theorem 2.22.

Let  and  be two ordered bases for a finite-dimensional vector space V, and let . Then

1. (a) Q is invertible.

2. (b) For any .

Proof.

1. (a) Since  is invertible, Q is invertible by Theorem 2.18 (p. 102).

2. (b) For any ,



by Theorem 2.14 (p. 92).

The matrix  defined in Theorem 2.22 is called a change of coordinate matrix. Because of part (b) of the theorem, we say that Q changes -coordinates into -coordinates. Observe that if  and , then



for ; that is, the jth column of Q is .

Notice that if Q changes -coordinates into -coordinates, then  changes -coordinates into -coordinates. (See Exercise 11.)

# Example 1

In , let  and . Since



the matrix that changes -coordinates into -coordinates is



Thus, for instance,



For the remainder of this section, we consider only linear transformations that map a vector space V into itself. Such a linear transformation is called a linear operator on V. Suppose now that T is a linear operator on a finite- dimensional vector space V and that  and  are ordered bases for V. Then V can be represented by the matrices  and . What is the relationship between these matrices? The next theorem provides a simple answer using a change of coordinate matrix.

# Theorem 2.23.

Let T be a linear operator on a finite-dimensional vector space V, and let  and  be ordered bases for V. Suppose that Q is the change of coordinate matrix that changes -coordinates into -coordinates. Then



Proof.

Let I be the identity transformation on V. Then ; hence, by Theorem 2.11 (p. 89),



Therefore .

# Example 2

Let T be the linear operator on  defined by



and let  and  be the ordered bases in Example 1. The reader should verify that



In Example 1, we saw that the change of coordinate matrix that changes -coordinates into -coordinates is



and it is easily verified that



Hence, by Theorem 2.23,



To show that this is the correct matrix, we can verify that the image under T of each vector of  is the linear combination of the vectors of  with the entries of the corresponding column as its coefficients. For example, the image of the second vector in  is



Notice that the coefficients of the linear combination are the entries of the second column of .

It is often useful to apply Theorem 2.23 to compute , as the next example shows.

# Example 3

Recall the reflection about the x-axis in Example 3 of Section 2.1. The rule  is easy to obtain. We now derive the less obvious rule for the reflection T about the line . (See Figure 2.5.) We wish to find an expression for T(a, b) for any (a, b) in . Since T is linear, it is completely determined by its values on a basis for . Clearly,  and . Therefore if we let



then  is an ordered basis for  and



Let  be the standard ordered basis for , and let Q be the matrix that changes -coordinates into -coordinates. Then



and . We can solve this equation for  to obtain that . Because





Since  is the standard ordered basis, it follows that T is left-multiplication by . Thus for any (a, b) in , we have



A useful special case of Theorem 2.23 is contained in the next corollary, whose proof is left as an exercise.

# Corollary.

Let  and let  be an ordered basis for . Then , where Q is the  matrix whose jth column is the jth vector of .

# Example 4

Let



and let



which is an ordered basis for . Let Q be the  matrix whose jth column is the jth vector of . Then



So by the preceding corollary,



The relationship between the matrices  and  in Theorem 2.23 will be the subject of further study in Chapters 5, 6, and 7. At this time, however, we introduce the name for this relationship.

# Definition.

Let A and B be matrices in . We say that B is similar to A if there exists an invertible matrix Q such that .

Observe that the relation of similarity is an equivalence relation (see Exercise 9). So we need only say that A and B are similar.

Notice also that in this terminology Theorem 2.23 can be stated as follows: If T is a linear operator on a finite-dimensional vector space V, and if  and  are any ordered bases for V, then  is similar to .

Theorem 2.23 can be generalized to allow , where V is distinct from W. In this case, we can change bases in V as well as in W (see Exercise 8).

# Exercises

1. Label the following statements as true or false.

1. (a) Suppose that  and  are ordered bases for a vector space and Q is the change of coordinate matrix that changes -coordinates into -coordinates. Then the jth column of Q is .

2. (b) Every change of coordinate matrix is invertible.

3. (c) Let T be a linear operator on a finite-dimensional vector space V, let  and  be ordered bases for V, and let Q be the change of coordinate matrix that changes -coordinates into -coordinates. Then .

4. (d) The matrices  are called similar if  for some .

5. (e) Let T be a linear operator on a finite-dimensional vector space V. Then for any ordered bases  and  for V,  is similar to .

2. For each of the following pairs of ordered bases  and  for , find the change of coordinate matrix that changes -coordinates into -coordinates.

1. (a)  and 

2. (b)  and 

3. (c)  and 

4. (d)  and 

3. For each of the following pairs of ordered bases  and  for , find the change of coordinate matrix that changes -coordinates into -coordinates.

1. (a)  and 

2. (b)  and 

3. (c)  and 

4. (d)  and 

5. (e)  and 

6. (f)  and 

4. Let T be the linear operator on  defined by



let  be the standard ordered basis for , and let



Use Theorem 2.23 and the fact that



to find .

5. Let T be the linear operator on  defined by , the derivative of p(x). Let  and . Use Theorem 2.23 and the fact that



to find .

6. For each matrix A and ordered basis , find . Also, find an invertible matrix Q such that .

1. (a)  and 

2. (b)  and 

3. (c)  and 

4. (d)  and 

7. In , let L be the line , where . Find an expression for T(x, y), where

1. (a) T is the reflection of  about L.

2. (b) T is the projection on L along the line perpendicular to L. (See the definition of projection in the exercises of Section 2.1.)

8. Prove the following generalization of Theorem 2.23. Let  be a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W. Let  and  be ordered bases for V, and let  and  be ordered bases for W. Then , where Q is the matrix that changes -coordinates into -coordinates and P is the matrix that changes -coordinates into -coordinates.

9. Prove that “is similar to” is an equivalence relation on .

1. (a) Prove that if A and B are similar  matrices, then . Hint: Use Exercise 13 of Section 2.3.

2. (b) How would you define the trace of a linear operator on a finite- dimensional vector space? Justify that your definition is well- defined.

10. Let V be a finite-dimensional vector space with ordered bases , and .

1. (a) Prove that if Q and R are the change of coordinate matrices that change α-coordinates into -coordinates and -coordinates into -coordinates, respectively, then RQ is the change of coordinate matrix that changes -coordinates into -coordinates.

2. (b) Prove that if Q changes -coordinates into -coordinates, then  changes -coordinates into -coordinates.

11. Prove the corollary to Theorem 2.23.

12. Let V be a finite-dimensional vector space over a field F, and let  be an ordered basis for V. Let Q be an  invertible matrix with entries from F. Define



and set . Prove that  is a basis for V and hence that Q is the change of coordinate matrix changing -coordinates into -coordinates. Visit goo.gl/vsxsGH for a solution.

13. Prove the converse of Exercise 8: If A and B are each  matrices with entries from a field F, and if there exist invertible  and  matrices P and Q, respectively, such that , then there exist an n-dimensional vector space V and an m-dimensional vector space W (both over F), ordered bases  and  for V and  and  for W, and a linear transformation  such that



Hints: Let , and  and  be the standard ordered bases for  and , respectively. Now apply the results of Exercise 13 to obtain ordered bases  and  from  and  via Q and P, respectively.