Math 307

1

• Math 307
• Spring, 2003
• Hentzel
• Time 110-200 MWF
• Room 1324 Howe Hall
• Instructor Irvin Roy Hentzel
• Office 432 Carver
• Phone [phone]
• E-mail hentzel_at_iastate.edu
• http//www.math.iastate.edu/hentzel/class.307.ICN
• Text Linear Algebra With Applications,
• Second Edition Otto Bretscher

2

• Monday, Mar 3 Chapter 4.2 Page 164
  Problems 6,14,16
• Main Idea Dejavu! We are doing everything all
  over again.
• Key Words Kernel, Image, Linear Transformation,
  rank, nullity
• Goal We want to expand the ideas of Matrices
  into a broader context.

3

• Previous Assignment
• The Leontief Problem
• The production of the plants R, S, and T for some
  period of
• time is given below.
• R S T
  Consumer Total
• R 10 10 30 30
  80
• S 10 20 10 20
  60
• T 60 20 10 10
  100
• The Leontief input-output model for the open
  model is
• X AX D.

4

• (1) What is the matrix A?
• (2) Solve the equation X AXD for X
• 145
• when D 290
• 145

5

• A
• 10/80 10/60 30/100
• 10/80 20/60 10/100
• 60/80 20/60 10/100

6

• Check with original system
• A X D X
• 10/80 10/60 30/100 80 30 80
• 10/80 20/60 10/100 60 20 60
• 60/80 20/60 10/100100 10 100
• It checks.

7

• Calculating X the new system where consumer
• 145
• demands is 290
• 145
• AXDX
• D (I-A)X
• (I-A) -1 D X

8

• 
  -1
• 1 0 0 10/80 10/60 30/100 145
  616
• 0 1 0 - 10/80 20/60 10/100 290
  690
• 0 0 1 60/80 20/60 10/100 145
  930

9

• 
  -1
• 7 1 3
• - -(--) -(---)
• 8 6 10
• 
  
• 1 2 1
  145
• X -(-) -- -(---)
  290
• 8 3 10
  145
• 
  
• 3 1 9
• -(-) -(--) ---
• 4 3 10

10

• 272 24 104
• ----- ---- ----
• 145 29 145
• 18 54 12 145
• X ---- ---- ---- 290
  
• 29 29 29 145
• 52 40 54
• ---- ---- ----
• 29 29 29

11

• 616
• X 690
• 930

12

• Previous Assignment
• Page 157 Problem 4
• Which of the subsets of P2 given in Exercises 1
  through 5 are subspaces of P2? Find a basis for
  those that are subspaces.
• t1
• p(t) INT p(t) dt 0
• t0

13

• 
  t1
• It is closed under addition since if INT p(t) dt
  0
• t1
  t0
• and INT q(t) dt 0,
• t0
• t1 t1
  t1
• then INT(p(t) q(t)) dt INT p(t) dt INT
  q(t) dt
• t0 t0
  t0
• 0 0 0
• So p(t) q(t) is also in the subspace.

14

• It is closed under scalar multiplication since if
  c is a number and
• t1 t1
  t1
• INT p(t) dt 0, then INT c p(t) dt c INT p(t)
  dt
• t0 t0
  t0
• c 0 0 so
• c p(t) is also in the subspace.

15

• Now we want to find a basis of the subspace.
• 1,x,x2 of P2. If p(x) a bx cx2 is any
  element of p2,
• 
  x1
• then p(x) is in the space if axbx2 /2 c
  x3/3 0
• 
  x0
• That is, if and only if a b/2 c/3 0

16

• This is a linear system 1 1/2 1/3 0 in
  Row Canonical Form.
• a -1/2 -1/3
• b u 1 v 0
• c 0 1
• So the basis is -1/2 x, and -1/3 x2.

17

• Page 157 Problem 10
• Which of the subsets of R3x3 given in Exercises 6
  through 11 are subspaces of R3x3.
• 
  1
• The 3x3 matrices A such that vector 2 is in
  the kernel of A.
  3

18

• The set is closed under addition since if AV 0,
  and BV 0, then
• (AB)V AV BV 0 so AB is also in the set.
• The set is closed under scalar multiplication
  since if c is a number and AV 0, then
• (cA)V c(AV) c0 0 so cA is also in the
  subset.

19

• A basis of the space is
• a b c 1 0
• d e f 2 0
• g h i 3 0
• a 2 b 3 c 0
• d 2 e 3 f 0
• g 2 h 3 i 0

20

• x1 x2 x3 x4 x5 x6
• a b c d e f g h i RHS
• 1 2 3 0 0 0 0 0 0 0
• 0 0 0 1 2 3 0 0 0 0
• 0 0 0 0 0 0 1 2 3 0
• a -2 -3 0
  0 0 0
• b 1 0 0
  0 0 0
• c 0 1 0
  0 0 0
• d 0 0 -2
  -3 0 0
• e x1 0 x2 0 x3 1 x4 0
  x5 0x6 0
• f 0 0 0
  1 0 0
• g 0 0 0
  0 -2 -3
• h 0 0 0
  0 1 0
• i 0 0 0
  0 0 0

21

• -2 1 0 0 0 0 0 0 0
• 0 0 0 -2 1 0 0 0 0
• 0 0 0 0 0 0 -2 1 0
• -3 0 1 0 0 0 0 0 0
• 0 0 0 -3 0 1 0 0 0
• 0 0 0 0 0 0 -3 0 1

22

• Page 157 Problem 20
• Find a basis for each of the spaces in Exercises
  16 through 31 and determine its dimension.
• The space of all matrices A a b in R2x2
• such that ad 0. c d

23

• The basis written as 2x2 matrices is
• 0 1 0 0 -1 0
• 0 0 1 0 0 1
• The dimension is 3.

24

• New Material
• A linear transformation TV ? W requires
• (1) V and W to be vector spaces.
• (2) T(V1V2) T(V1)T(V2).
• (3) T(c V) c T(V).

25

• _________ ____________
• V
  W
• iiiii
  
• kk T iiiiiiii
  
• kkkk ------------gt0iiiiii
  
• kkk iiii
  
• ii
  
• _________ ____________

26

• Points of interest.
• (4) Ker(T) V T(V) 0.
• (5) Im(T) T(V) V is in V
• (6) T is invertible is equivalent to
• (a) T(V) 0 gt V 0. Ker(T) 0
• (b) T(X) W is solvable for every W.
  Im(T)W

27

• Main Theorem
• Dim(V) Dim( IMAGE ) Dim(KERNEL).
• Pretty much that is what you would expect. The
  quantity you started with is how much you have
  left plus how much was lost.

28

• We now discuss the kernel and the range for
  various functions.

29

• Example 1.
• For differentiation, what is the kernel?

30

• The kernel is the set of all functions which map
  to zero.
• That is, those functions whose derivative is 0.
• Those are the constant functions like
• y constant.

31

• The range is the set of all functions which are
  derivatives of something.
• This includes all continuous functions.

32

• Is differentiation invertible?

33

• No because you cannot recover the constant.
• Both y x2 and y x21 have the same
  derivative.
• Thus having been given only y 2x, you cannot
• determine the original function.
• Thus differentiation does not have an inverse.

34

• Example 2.
• Is integration invertible?

35

• Let INT f be the indefinite integral of f.
• INT fg INT f INT g
• INT cf c INT f
• So integration satisfies the requirements to be a
  linear transformation.

36

• There is one small problem. INT is not a
  function since as it stands, there is not one,
  but many possible integrals.
• We can patch up this problem by saying INT f is
  the function F such that F' f and F0 0.

37

• Notice that d/dx INT f f so that INT has a
  left inverse.
• However INT does not have a right inverse since
  INT d/dx f will only equal f(x) - f(0).
• Since you do not get f(x) back exactly,
  integration does not have an inverse.

38

• We once remarked that if a matrix has a left
  inverse, it also has a right inverse and the two
  inverses are equal.

39

• This theorem requires that the matrix be square
  and the space be finite dimensional.
• It is not true for non square matrices nor for
  infinite dimensional spaces.

40

• Example 3.
• Consider the function R2x2 ----gt R2x2 given by
• a b ? 1 2 a b
• c d 2 4 c d
• Is this function invertible?

41

• We see if anything is mapped to zero.
• That is, is there any matrix a b
• c
  d
• such that 1 2 a b 0 ?
• 2 4 c d

42

• Thus a2c 0 b2d 0 2a4c 0 2b4d 0
• a b c d RHS a
  b cu dv RHS
• 1 0 2 0 0 1
  0 2 0 0
• 0 1 0 2 0 0
  1 0 2 0
• 2 0 4 0 0 0
  0 0 0 0
• 0 2 0 4 0 0
  0 0 0 0
• a -2 0
• b u 0 v -2
• c 1 0
• d 0 1

43

• The kernel contains -2u -2v
• u v
• So since some information is lost, the function
  is not invertible.

44

• Define TX AX-XA for a fixed matrix A.
• 1. Show that T is a linear transformation.
• 2. Show that T is never invertible.

45

• We have to first show that TXY TXTY.

46

• TXY A(XY) (X Y)A
• AXAY XA YA
• AX XA AY YA
• TX TY.

47

• TcX A(cX) (cX)A
• c(AXXA)
• cTX.

48

• The kernel of T contains the identity matrix I so
  T is not invertible.

49

• What is the matrix for T if A 1 2
• 
  3 4

50

• 
  1 0 0 1 0 0 0 0
• 
  0 0 0 0 1 0 0 1
• 1 0 ? 1 2 - 1 0 0 2 0
  2 -3 0
• 0 0 0 0 3 0 -3 0
  
• 
  
• 0 1 ? 3 4 - 0 1 3 3 3
  3 0 -3
• 0 0 0 0 0 3 0-3
  
• 
  
• 0 0 ? 0 0 - 2 0 -2 0 -2 0
  -3 2
• 1 0 1 2 4 0 -3 2
  
• 
  
• 0 0 ? 0 0 - 0 2 0 -2 0 -2
  3 0
• 0 1 3 4 0 4 3 0
  

51

• The matrix for T is
• 0 3 -2 0
• 2 3 0 -2
• -3 0 -3 3
• 0 -3 2 0

52

• What is the kernel of T?

53

• 0 3 -2 0 -3 0 -3 3 1 0 1 -1
• 2 3 0 -2 0 3 -2 0 0 3 -2 0
• -3 0 -3 3 2 3 0 -2 2 3 0 -2
• 0 -3 2 0 0 -3 2 0 0 -3 2 0
• 1 0 1 -1 1 0 1 -1 1 0 1 -1
  
• 0 3 -2 0 0 3 -2 0 0 1 -2/3 0
• 0 3 -2 0 0 0 0 0 0 0 0 0
  
• 0 -3 2 0 0 0 0 0 0 0 0 0

54

• x y za wb
• 1 0 1 -1
• 0 1 -2/3 0
• 0 0 0 0
• 0 0 0 0
• The kernel is
• x -1 1
• y a 2/3 b 0
• z 1 0
• w 0 1

55

• So the elements in the kernel are linear
  combinations of
• -1 2/3 and 1 0
• 1 0 0 1
• I.E. linear combinations of A and I.

56

• Example 4.
• Suppose we have a linear transformation T
• which reflects across a line. What is the
  matrix
• of T with respect to the standard basis?
• 1 0
• 0 1

57

• Case 1. /\ /
• / y x
• /
• /
• --------------------------------?--------------
  
• /
• /
• /

58

• Case 2. \ /\
• \ y -x
• \
• --------------------------------?--------------
• \
• \