Calculus 6

Chapter Six

Linear Functions and Matrices 6.1 Matrices Suppose f : R n → R p be a linear function. Let e1 , e 2 ,K , e n be the coordinate vectors for R n . For any x ∈ R n , we have x = x 1e 1 + x 2 e 2 +K+ x n e n . Thus f ( x) = f ( x1 e 1 + x 2 e2 +K+ x n e n ) = x1 f ( e1 ) + x 2 f ( e2 )+K+ xn f (e n ) .

Meditate on this; it says that a linear function is entirely determined by its values f ( e1 ), f (e 2 ),K , f (e n ) . Specifically, suppose f ( e1 ) = (a11 , a21 , K, a p1 ), f ( e 2 ) = (a12 , a 22 ,K , a p 2 ), M f ( e n ) = (a1 n , a 2 n ,K , a pn ).

Then f ( x) = ( a11 x1 + a12 x 2 +K+ a1n x n , a 21 x1 + a 22 x 2 +K+ a 2 n x n ,K , a p 1 x 1 + a p 2 x 2 +K+ a pn x n ).

The numbers aij thus tell us everything about the linear function f. . To avoid labeling these numbers, we arrange them in a rectangular array, called a matrix:

6.1

 a11 a12 K a1 n  a   21 a 22 K a2 n   M M     a p 1 a p 2 K a pn 

The matrix is said to represent the linear function f. For example, suppose f : R 2 → R 3 is given by the receipt

f ( x1 , x 2 ) = (2 x1 − x 2 , x1 + 5x 2 , 3x 1 − 2 x2 ) .

Then f ( e1 ) = f (10 , ) = ( 2,1,3) , and f ( e 2 ) = f (01 , ) = ( −15 , ,− 2) . The matrix representing f is thus  2 −1 1 5     3 − 2

Given the matrix of a linear function, we can use the matrix to compute f ( x) for any x. This calculation is systematized by introducing an arithmetic of matrices. First, we need some jargon. For the matrix

 a11 a 21 A=  M   a p1

a12 a22 a p2

6.2

K a1n  K a 2 n  , M   K a pn 

the matrices [ai 1 , a i2 ,K , ain ]

 a1 j  a  2j are called rows of A, and the matrices   are called  M     a pj 

columns of A. Thus A has p rows and n columns; the size of A is said to be p × n . A vector in R n can be displayed as a matrix in the obvious way, either as a 1× n matrix, in which case the matrix is called a row vector, or as a n × 1 matrix, called a column vector. Thus the matrix representation of f is simply the matrix whose columns are the column vectors f ( e1 ), f (e1 ),K , f (e n ) .

Example Suppose f : R 3 → R 2 is defined by

f ( x 1 , x 2 , x 3 ) = (2 x1 − 3 x2 + x 3 , − x 1 + 2 x2 − 5x 3 ) .

So f ( e1 ) = f (10 , ,0) = (2,− 1) , f ( e 2 ) = f (0,1,0 ) = ( −32 , ) , and f ( e 3 ) = f ( 0,0 ,1) = (1,− 5) . The matrix which represents f is thus  2 −3 1   −1 2 − 5  

Now the recipe for computing f(x) can be systematized by defining the product of a matrix A and a column vector x. Suppose A is a p × n matrix and x is a n × 1 column

6.3

vector. For each i = 12 , , K, p, let ri denote the i th row of A . We define the product Ax to be the p × 1column vector given by  r1 ⋅ x  r ⋅ x  2 . Ax =   M     rp ⋅ x 

If we consider all vectors to be represented by column vectors, then a linear function f : R n → R p is given by f ( x) = Ax , where, of course, A is the matrix representation of f.

Example Consider the preceding example: f ( x 1 , x 2 , x 3 ) = (2 x1 − 3 x2 + x 3 , − x 1 + 2 x2 − 5x 3 ) . We found the matrix representing f to be  2 −3 1  A= .  −1 2 −5 Then  x1   2 − 3 1     2x 1 − 3x 2 + x 3  Ax =   x2 =   = f ( x)  − 1 2 − 5  x   − x1 + 2 x 2 − 5x 3   3

Exercises

6.4

1. Find the matrix representation of each of the following linear functions: a) f ( x1 , x 2 ) = (2 x1 − x 2 , x1 + 4 x 2 , -7x 1 , 3x1 + 5x 2 ) . b) R (t ) = 4t i − 5tj − 2t k . c) L( x ) = 6x .

2  −2 2. Let g be define by g( x ) = Ax , where A =  0  3

−1 1  . Find g(3,−9 ) . −3  5

3. Let f : R 2 → R 2 be the function in which f(x) is the vector that results from rotating the vector x about the origin

π in the counterclockwise direction. 4

a)Explain why f is a linear function. b)Find the matrix representation for f. d)Find f(4,-9).

4. Let f : R 2 → R 2 be the function in which f(x) is the vector that results from rotating the vector x about the origin θ in the counterclockwise direction. Find the matrix representation for f.

5. Suppose g: R 2 → R 2 is a linear function such that g(1,2) = (4,7) and g(-2,1) = (2,2).

6.5

Find the matrix representation of g.

6. Suppose f : R n → R p and g: R p → R q are linear functions.

Prove that the

composition g o f : R n → R q is a linear function.

7. Suppose f : R n → R p and g: R n → R p are linear functions. Prove that the function f + g: R n → R p defined by ( f + g)( x ) = f ( x ) + g ( x) is a linear function.

6.2 Matrix Algebra Let us consider the composition h = g o f of two linear functions f : R n → R p and g: R p → R q . Suppose A is the matrix of f and B is the matrix of g. Let’s see about the matrix C of h. We know the columns of C are the vectors g ( f (e j )), j = 12 , ,K, n , where, of course, the vectors e j are the coordinate vectors for R n . Now the columns of A are just the vectors f ( e j ), j = 1,2,K , n . Thus the vectors g ( f (e j )) are simply the products Bf (e j ) . In other words, if we denote the columns of A by ki , i = 12 , ,K, n , so that A = [k1 , k 2 ,K , k n ] , then the columns of C are Bk1 , Bk2 ,K, Bkn , or in other words, C = [ Bk1 , Bk2 ,K, Bkn ] .

Example

6.6

2 1 0 −1 −5 8   and let the matrix A of f be Let the matrix B of g be given by B =   2 7 −3    2 −2 1  3 given by A =  1  −4

1 2  . −3

Thus

f : R 2 → R 3 and g: R 3 → R 4 (Note that for the

composition h = g o f to be defined, it must be true that the number of columns of B be  3 1   the same as the number of rows of A.). Now, k1 =  1  and k2 =  2  , and so  − 4  −3  −5   −5   −40   −35 Bk1 =  and Bk = 2  25  . The matrix C of the composition is thus 25   −3   0   −5 −5  −40 −35  C= . 25 25   0 −3 

These results inspire us to define a product of matrices. Thus, if B is an n × p matrix, and A is a p × q matrix, the product BA of these matrices is defined to be the

n × q matrix whose columns are the column vectors Bk j , where k j is the j th column of A. Now we can simply say that the matrix representation of the composition of two linear functions is the product of the matrices representing the two functions.

6.7

There are several interesting and important things to note regarding matrix products. First and foremost is the fact that in general BA ≠ AB , even when both products are defined (The product BA obviously defined only when the number of columns of B is the same as the number of rows of A.). Next, note that it follows directly from the fact that h o ( f o g ) = (h o f ) o g that for C(BA) = (CB)A. Since it does not matter where we insert the parentheses in a product of three or more matrices, we usually omit them entirely. It should be clear that if f and g are both functions from R n to R p , then the matrix representation for the sum f + g: R n → R p is simply the matrix  a11 + b11 a + b 21 21 A+ B =   M   a p 1 + bp 1

a12 + b12 a22 + b22 a p2 + b p2

K a1 n + b1 n  K a 2 n + b2 n  ,   K a pn + b pn 

where  a11 a 21 A=  M  a p1

a12 a22 a p2

is the matrix of f, and

6.8

L a1n  L a 2 n    L a pn 

 b11 b12 L b1n  b b22 L b2 n  21  B=  M    b p 1 b p 2 L bpn 

is the matrix of g. Meditating on the properties of linear functions should convince you that for any three matrices (of the appropriate sizes) A, B, and C, it is true that

A( B + C ) = AB + AC .

Similarly, for appropriately sized matrices, we have ( A + B)C = AC + BC .

Exercises 8. Find the products:  2 1  −2  a)     0 3  1 

 2 1 1 b)     0 3 3

 2 1  −2 1 c)     0 3  1 3

1 5  −2 3  d) [1 −3 2 − 1] 0 2    −3 4

 1 0 0 a 11 9. Find a)  0 1 0 a 21  0 0 1 a 31

a12 a 22 a 32

a13  a 23  a 33 

 0 0 0 a 11 b)  0 0 0 a 21  0 0 0 a 31

6.9

a12 a 22 a 32

a13  a 23  a 33 

10. Let A(θ ) be the 2 × 2 matrix for the linear function that rotates the plane θ counterclockwise. Compute the product A(θ ) A(η ) , and use the result to give identities for cos(θ + η ) and sin(θ + η ) in terms of cosθ , cosη , sinθ , and sinη .

11. a)Find the matrix for the linear function that rotates R 3 about the coordinate vector j by

π (In the positive direction, according to the usual “right hand rule” for rotation.). 4

b)Find a vector description for the curve that results from applying the linear transformation in a) to the curve R (t ) = cos ti + sin tj + t k .

12. Suppose f : R 2 → R 2 is linear. Let C be the circle of radius 1 and center at the origin. Find a vector description for the curve f(C).

13. Suppose g: R 2 → R n is linear.

Suppose moreover that

g( −11 , ) = ( 4,− 5) . Find the matrix of g.

6.10

g(11 , ) = (2,3)

and