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 2xex2 can be found by making the change of variable u=x2. The resulting expression is of such a simple form that an antiderivative is easily recognized:

∫2xex2dx=∫eudu=eu+c=ex2+c.

Similarly, in geometry the change of variable

x=25x′−15y′y=15x′+25y′

can be used to transform the equation 2x2−4xy+5y2=1 into the simpler equation (x′)2+6(y′)2=1, 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

(xy)→(x′y′)
The graph of an ellipse and 2 vectors.

Figure 2.4

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 x′y′-coordinate system with coordinate axes rotated from the original xy-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 x′-axis and the y′-axis form an ordered basis

β′={15(21), 15(−12)}

for R2, and the change of variable is actually a change from [P]β=(xy), the coordinate vector of P relative to the standard ordered basis β={e1, e2}, to [P]β′=(x′y′), the coordinate vector of P relative to the new rotated basis β′.

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

(xy)=15(2−112)(x′y′).

Notice also that the matrix

Q=15(2−112)

equals [I]β′β, where I denotes the identity transformation on R2. Thus [v]β=Q[v]β′ for all v∈R2. A similar result is true in general.

Theorem 2.22.

Let β and β′ be two ordered bases for a finite-dimensional vector space V, and let Q=[IV]β′β. Then

  1. (a) Q is invertible.

  2. (b) For any v∈V, [v]β=Q[v]β′.

Proof.

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

  2. (b) For any v∈V,

    [v]β=[IV(v)]β=[IV]β′β[v]β′=Q[v]β′

    by Theorem 2.14 (p. 92).

The matrix Q=[IV]β′β defined in Theorem 2.22 is called a change of coordinate matrix. Because of part (b) of the theorem, we say that Q changes β′-coordinates into β-coordinates. Observe that if β={x1, x2, …, xn} and β′={x′1, x′2, …, x′n}, then

x′j=∑i=1nQijxi

for j=1, 2, …, n; that is, the jth column of Q is [x′j]β.

Notice that if Q changes β′-coordinates into β-coordinates, then Q−1 changes β-coordinates into β′-coordinates. (See Exercise 11.)

Example 1

In R2, let β={(1, 1), (1, −1)} and β′={(2, 4), (3, 1)}. Since

(2, 4)=3(1, 1)−1(1, −1)      and      (3, 1)=2(1, 1)+1(1, −1),

the matrix that changes β′-coordinates into β-coordinates is

Q=(32−11).

Thus, for instance,

[(2, 4)]β=Q[(2, 4)]β′=Q(10)=(3−1).

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 β and β′ are ordered bases for V. Then V can be represented by the matrices [T]β and [T]β′. 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 β and β′ be ordered bases for V. Suppose that Q is the change of coordinate matrix that changes β′-coordinates into β-coordinates. Then

[T]β′=Q−1[T]βQ.

Proof.

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

Q[T]β′=[I]β′β[T]β′β′=[IT]β′β=[TI]β′β=[T]ββ[I]β′β=[T]βQ.

Therefore [T]β′=Q−1[T]βQ.

Example 2

Let T be the linear operator on R2 defined by

T(ab)=(3a−ba+3b),

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

[T]β=(31−13).

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

Q=(32−11),

and it is easily verified that

Q−1=15(1−213).

Hence, by Theorem 2.23,

[T]β′=Q−1[T]βQ=(41−22).

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

T(31)=(86)=1(24)+2(31).

Notice that the coefficients of the linear combination are the entries of the second column of [T]β′.

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

Example 3

Recall the reflection about the x-axis in Example 3 of Section 2.1. The rule (x, y)→(x, −y) is easy to obtain. We now derive the less obvious rule for the reflection T about the line y=2x. (See Figure 2.5.) We wish to find an expression for T(a, b) for any (a, b) in R2. Since T is linear, it is completely determined by its values on a basis for R2. Clearly, T(1, 2)=(1, 2) and T(−2, 1)=−(−2, 1)=(2, −1). Therefore if we let

β′={(12), (−21)},
A graph of the reflection T where y equals 2 x.

Figure 2.5

then β′ is an ordered basis for R2 and

[T]β′=(100−1).

Let β be the standard ordered basis for R2, and let Q be the matrix that changes β′-coordinates into β-coordinates. Then

Q=(1−221)

and Q−1[T]βQ=[T]β′. We can solve this equation for [T]β to obtain that [T]β=Q[T]β′Q−1. Because

Q−1=15(12−21),

the reader can verify that

[T]β=15(−3443).

Since β is the standard ordered basis, it follows that T is left-multiplication by [T]β. Thus for any (a, b) in R2, we have

T(ab)=15(−3443)(ab)=15(−3a+4b4a+3b).

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

Corollary.

Let A∈Mn×n(F) and let γ be an ordered basis for Fn. Then [LA]γ=Q−1AQ, where Q is the n×n matrix whose jth column is the jth vector of γ.

Example 4

Let

A=(2101130−10),

and let

γ={(−100), (210), (111)},

which is an ordered basis for R3. Let Q be the 3×3 matrix whose jth column is the jth vector of γ. Then

Q=(−121011001)       and      Q−1=(−12−101−1001).

So by the preceding corollary,

[LA]γ=Q−1AQ=(028−1460−1−1).

The relationship between the matrices [T]β′ and [T]β 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 Mn×n(F). We say that B is similar to A if there exists an invertible matrix Q such that B=Q−1AQ.

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 β and β′ are any ordered bases for V, then [T]β′ is similar to [T]β.

Theorem 2.23 can be generalized to allow T:V→W, 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 β={x1, x2, …, xn} and β′={x′1, x′2, …, x′n} are ordered bases for a vector space and Q is the change of coordinate matrix that changes β′-coordinates into β-coordinates. Then the jth column of Q is [xj]β′.

    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 β and β′ be ordered bases for V, and let Q be the change of coordinate matrix that changes β′-coordinates into β-coordinates. Then [T]β=Q[T]β′Q−1.

    4. (d) The matrices A, B∈Mn×n(F) are called similar if B=QtAQ for some Q∈Mn×n(F).

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

  2. For each of the following pairs of ordered bases β and β′ for R2, find the change of coordinate matrix that changes β′-coordinates into β-coordinates.

    1. (a) β={e1, e2} and β′={(a1, a2), (b1, b2)}

    2. (b) β={(−1, 3), (2, −1)} and β′={(0, 10), (5, 0)}

    3. (c) β={(2, 5), (−1, −3)} and β′={e1, e2}

    4. (d) β={(−4, 3), (2, −1)} and β′={(2, 1), (−4, 1)}

  3. For each of the following pairs of ordered bases β and β′ for P2(R), find the change of coordinate matrix that changes β′-coordinates into β-coordinates.

    1. (a) β={x2, x, 1} and β′={a2x2+a1x+a0, b2x2+b1x+b0, c2x2+c1x+c0}

    2. (b) β={1, x, x2} and β′={a2x2+a1x+a0, b2x2+b1x+b0, c2x2+c1x+c0}

    3. (c) β={2x2−x, 3x2+1, x2} and β′={1, x, x2}

    4. (d) β={x2−x+1, x+1, x2+1} and β′={x2+x+4, 4x2−3x+2, 2x2+3}

    5. (e) β={x2−x, x2+1, x−1} and β′={5x2−2x−3, −2x2+5x+5, 2x2−x−3}

    6. (f) β={2x2−x+1, x2+3x−2, −x2+2x+1} and β′={9x−9, x2+21x−2, 3x2+5x+2}

  4. Let T be the linear operator on R2 defined by

    T(ab)=(2a+ba−3b),

    let β be the standard ordered basis for R2, and let

    β′={(11), (12)}.

    Use Theorem 2.23 and the fact that

    (1112)−1=(2−1−11)

    to find [T]β′.

  5. Let T be the linear operator on P1(R) defined by T(p(x))=p′(x), the derivative of p(x). Let β={1, x} and β′={1+x, 1−x}. Use Theorem 2.23 and the fact that

    (111−1)−1=(121212−12)

    to find [T]β′.

  6. For each matrix A and ordered basis β, find [LA]β. Also, find an invertible matrix Q such that [LA]β=Q−1AQ.

    1. (a) A=(1311) and β={(11), (12)}

    2. (b) A=(1221) and β={(11), (1−1)}

    3. (c) A=(11−1201110) and β={(111), (101), (112)}

    4. (d) A=(131411344410) and β={(11−2), (1−10), (111)}

  7. In R2, let L be the line y=mx, where m≠0. Find an expression for T(x, y), where

    1. (a) T is the reflection of R2 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 T:V→W be a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W. Let β and β′ be ordered bases for V, and let γ and γ′ be ordered bases for W. Then [T]β′γ′=P−1[T]βγQ, where Q is the matrix that changes β′-coordinates into β-coordinates and P is the matrix that changes γ′-coordinates into γ-coordinates.

  9. Prove that “is similar to” is an equivalence relation on Mn×n(F).

    1. (a) Prove that if A and B are similar n×n matrices, then tr(A)=tr(B). 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.

  11. Let V be a finite-dimensional vector space with ordered bases α, β, and γ.

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

    2. (b) Prove that if Q changes α-coordinates into β-coordinates, then Q−1 changes β-coordinates into α-coordinates.

  12. Prove the corollary to Theorem 2.23.

  13. † Let V be a finite-dimensional vector space over a field F, and let β={x1, x2, …, xn} be an ordered basis for V. Let Q be an n×n invertible matrix with entries from F. Define

    x′j=∑i=1nQijxi      for  1≤j≤n,

    and set β′={x′1, x′2, …, x′n}. Prove that β′ is a basis for V and hence that Q is the change of coordinate matrix changing β′-coordinates into β-coordinates. Visit goo.gl/vsxsGH for a solution.

  14. Prove the converse of Exercise 8: If A and B are each m×n matrices with entries from a field F, and if there exist invertible m×m and n×n matrices P and Q, respectively, such that B=P−1AQ, then there exist an n-dimensional vector space V and an m-dimensional vector space W (both over F), ordered bases β and β′ for V and γ and γ′ for W, and a linear transformation T:V→W such that

    A=[T]βγ     and      B=[T]β′γ′.

    Hints: Let V=Fn, W=Fm, T=LA, and β and γ be the standard ordered bases for Fn and Fm, respectively. Now apply the results of Exercise 13 to obtain ordered bases β′ and γ′ from β and γ via Q and P, respectively.

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