I 1 I 1 - University of California, Davis

S Chapter 1 Systems of Linear Equations and Matrices

Add --3 times the first equation to the third to obtain

x+ y+ 2z= 9 2y-- 7z=--17 -- liz = --27

Add --3 times the first row to the third to obtain

11

2

9

o 2 --7 --17

o 3 --11 --27

Multiply the second equation by to obtain

x+

y+ 2z= 9

-- ` 7__ I?

--

-- liz = --27

Multiply the second row by to obtain

II

2

9

II

i

7

U

2

2

0 3 --11 --27

Add --3 times the second equation to the third to obtain

x+y+2z= 9

7

y

--

--

I..

3

5' --

--

_5

Multiply the third equation by --2 to obtain

x+y+2z= 9

-- ` 7_-- _ 17

z= 3

Add --3 times the second row to the third to obtain

I12 9

o

7

17

2

2

o

o--

--

MultipI the third row by --2 to obtain

II2 9

0

i

7

7

2

2

oo 1 3

Add --l times the second equation to the first to obtain

-- 35

X

T'

7

Y

7,. 2'

17 2

2= 3

Add--I times the second row to the first to obtain

II

35

o

i

17

2

2

o0 1 3

The solution in this example can also be expressed as the or dered triple (1,2,3) with the understanding that the num bers in the triple are in the same order as the variables in the system, namely, .r, y, z.

Add times the third equation to the first --

and times the third equation to the second to

obtain

x

=1

y =2 z=3

Add -- times the third row to the first and times the third row to the second to obtain

1 00 1

o 102

o0I 3

The solution x = 1. y = 2, z = 3 is now evident. I

ExerCise Set 1.1

I. In each part, determine whether the equation is linear in x1, i, and x3.

(a) x1 + 5x3 -- V'5X1 = I

(b) x1 + 3x2 + X1X3 = 2

(c) X1 = --7x2 + 3x3

(d) Xj +X2 + Ru = 5

(e) X' --2x---X =4

(1) ni -- i5'x1 = 71/3

2. in each part, determine whether the equation is linear in x and y.

(a) 2113x + s/Iy =

(b) 2XV + 3/7 = I

(c) cos (t)X -- = Iog3

(d) cosu--4y=O

(e) xy =

(1) y -- 7 = x

r

4

r

1.1 Introduction to Systems of Linear Equations 9

3. UsingthenotationofFormula(7), writedownagenerallinear system of

(a) two equations in two unknowns.

(b) three equations in three unknowns.

(c) two equations in four unknowns,

4. Write down the augmented matrix for each of the linear systems in Exercise 3.

In each part of Exercises - 6, find a linear system in the unknowns x1 ? x2, x3,..., that corresponds to the given augmented

matrix.

r2

ol

F3 0--2 s1

5. (a) 3 --4 0

i ij

(b) 7 1

[o --2 I 7j

ro 6. (a) ,5

r

1--4

(b) I_I

[0

3 --l --l --I] 2 0 --3 _6j

o I --4 3]

0 4 I --31 3 0 2 --91

0 0 --I

In each part of Exercises 7--H, find the augmented matrix for the linear system.

7. (a) --Zr1= 6 3x1= 8 9x1 = --3

(b)6x1 -- x'+3x3=4 It2-- x3l

(c)

2x2

--3x4+ X5= 0

--it1 -- x2+xj 6x1 + 2x2 -- x3 + 2X4 -- 3x5 = 6

8. (a) it1 -- Zr' = --l 4x1 + It = 3 Th1 +3x2 = 2

(b) 2x1

+ Zr3 = I

it1 -- x2 + 4x3 = 7

6x1 +X2 -- .t, =0

(c) x1

=1 =2 X3 =3

9. In each part, determine whether the given 3-tupleis a solution ofthelinearsystem

Zr1 --4x2 -- x1 = I -- 3x2 + .r3 = 1

Zr1 -- Sx -- Zr3 = I

(a) (3,1,1)

(b) (3, --1,1)

(c) (13,5,2)

(d) (#,,2)

(e) (I7,7,5)

10. In each part, determine whether the given 3-tuple is a solution of the linear system x1-2y--2z3 5X-- y+ ZI --x?5y--5:='S

(a) (, [ I)

, (b) (, 0)

(c) (5,8, I)

(d) (5

10 2)

--.

(e) (55, --

11. In each part, solve the linear system, if possible, and use the result to determine vhether the lines represented by the equa tions in the system have zero, one, or infinitely many points of intersection. If there is a single point of intersection, give its coordinates, and if there are infinitely many, find parametric

equations for them.

(a) Zr 2y = 4 -- 6x--4y=9

(b) Zr 4y = I -- 4x--8y=2

(c) x -- 2y = 0 x--4y=8

12. Under what conditions on a and b will the following linear system have no solutions, one solution, infinitely many solu

tions?

it

=a

-

4x--6y=b

In each part of Exercises 13--13, use parametric equations to describe the solution set of the linear equation. `4

13(a) 7x--5y3 (b)3x1 --5x,+4x37 (c) --8x1 + Zr, -- 5i + 6x. = 1 (d) 3v -- 8w ? Zr y + 4z = 0

--

14. (a) x + IGy = 2

(b)xj+3x2--12x23

(c) 4x1 +2x, + Zr3

= 20

(d)v+w+x--5y+7z=0

In Exercises 15--16, each linear system has infinitely many so lutions. Use parametric equations to describe its solution set.

15. (a) 2x 3y = -- 6x -- 9v = 3

(b) x1--3x1-- x3= --4 it1 ?9x2--3x3--12 --x1--3x'+ x= 4

16 (a) 6x1 ? Zr1 = --s

3x1+ x=--4

(b) Zr y + 2z = --4 -- 6x--3y-,-6:=--12 --4.r+2v--4z= 8

In Exercises 17--Ill, find a single elementary row operation that will create a I in the upper left corner of the given augmented ma trix and will not create any fractions in its first row.

r-- --l 2 1

Fo --l --s o]

17.(a) 2 --3 3 2 (b) 12 --9 3 2

L o 2 --3 lJ

[1 4 --3 3J

F2

l8,(a) 7

[--s

4 --6 I4 42

81

7J

r --4 --2 I(b) 3 1 8

L--6 3 --l

21

4IJi

10 Chapter 1 Systems of Linear Equations and Matrices

In Exercises 19--20, find all values of k for which the given auQmented matrix corresponds to a consistent linear system.

Lets, y, and z denote the number of ounces of the ftrsi, sec ond, and third foods that the dieter will consume at the main

I,

19. (a)

1]

(b)

k --l 8 --4

meal. Find (but do not solve) a linear system in s, y, and z whose solution tells how many ounces of each food must be consumed to meet the diet requirements.

[1 20. (a)

--4 k 85

(b)

I --2 --I 2

21. The curve y = as2 ? bx + r shown in the accompanying fig ure passes through the points (x1, yl), (x2, Y2), and (a3, y). Show that the coefficients a, b, and c form a solution of the system of linear equations whose augmented matrix is

.t1 I Yt

x I Y2

4 53 1 y3

26. Suppose that you want to find values for a, b, and c such that the parabola y = ax + bx + c passes through the points (I, I). (2,4), and (--1,1). Find (but do not solve) a system of linear equations whose solutions provide values for a, and c. How many solutions would you expect this system of equations to have, and why?

27. Suppose you are asked to find three real numbers such that the sum of the numbers is 12, the sum of two times the first plus the second plus Iwo times the third is 5, and the third number is one more than the first. Find (but do not solve) a linear system whose equations describe the three conditions.

= 052 hx + c

True-False Exercises

TF- In parts (a)--(h) determine whether the statement is true or false, and justify your answer.

(a) A linear system whose equations are all homogeneous must be consistent.

4 Figure Ex-21

(b) Multiplying a ro'v of an augmented matrix through by zero is an acceptahlc elementary row operation.

22. Explain why each of the three elementary row operations does not affect the solution set of a linear system.

23. Show that if the linear equations

t -s-ks2 =c and Si +1x2 =d have the same solution set, then the two equations are identical (i.e., k = I and c = d).

24. Consider the system of equations

ax + by = k cx + dy = I ex + fy =

Discuss the relative positions of the lines ax + by = cx + dy = 1, andes + fy = in when (a) the system has no solutions.

(b) the system has exactly one solution.

(c) the 5 stem has infinitely many solutions.

25. Suppose that a certain diet calls for 7 units of fat. 9 units of protein, and 16 units of carbohydrates for the main meal, and suppose that an individual has three possible foods to choose from to meet these requirements: Food I: Each ounce contains 2 units of fat, 2 units of protein, and 4 units of carbohydrates. Food 2: Each ounce contains 3 units of fat, I unit of protein, and 2 units of carbohydrates. Food 3: Each ounce contains I unit of fat, 3 units of protein, and 5 units of carbohydrates.

(c) The linear system 5-- y=3

2x 2y = k --

cannot have a unique solution, regardless of the value of k.

(d) A single linear equation with two or more unknowns must have infinitely many solutions.

(e) If the number of equations in a linear system exceeds the num ber of unknowns then the system must be inconsistent.

(f) If each equation in a consistent linear system is multiplied through by a constant c, then all solutions to the new system can be obtained by multiplying solutions from the original syslem by c.

(g) Elementary row operations permit one row of an augmented matrix to be subtracted from another.

(h) The linear system with corresponding augmented matrix

r2 --l 4

[o 0 --l

is consistent.

Working with Technology TI. Solve the linear systems in Examples 2. 3, and 4 to see how your technology utility handles the three types of systems.

1'2. Use the result in Exercise 21 to find values of a, b, and r for which the curve ` = ax2 + bx + c passes through the points (--1.1,4). (0,0, 8), and (I, 1.7).

22 chapter 1 Systems of Linear Equations and Matrices

If A is the augmented ma nix for a linear system, then the pivot columns identify the leading variables. As an illus tration, in Example S the pivot columns are I, 3, and 6, and the leading variables arex1 , and x5.

1 EXAMPLE 9 Pivot Positions and Columns Earlier in this section (immediately after Definition 1) we found a row echelon form of

0 A= 2

2

0 --2 4 --lO 4 --5

0 7 12 6 12 28 6 --5 --l

to be

1 2 --5 3 6 14

0

0

I

0 -- --6

0000 1 2

The leading Vs occur in positions (row 1, column I), (row 2, column 3), and (row 3, column 5). These arc the pivot positions. The pivot columns are columns 1, 3, and 5.

.4

Roundoff Error and Instability

There is often a gap between mathematical theory and its practical implementation-- Gauss--Jordan elimination and Gaussian elimination being good examples. The problem is that computers generally approximate numbers, thereby introducing roundoff errors, so unless precautions are taken, successive calculations may degrade an answer to a degree that makes it useless. Algorithms (procedures) in which this happens are called unstable. There are various techniques for minimizing roundoff error and instability For example, it can be shown that for large linear systems Gauss--Jordan elimination involves roughly 50% more operations than Gaussian elimination, so most computer algorithms are based on the latter method. Some of these matters will be considered in Chapter 9.

Exercise Set 1.2

In Exercises 1--2, determine whether the matrix is in row ech elon form, reduced row echelon form, both, or neither.

1.(a)

1 00 010 001

1 00 (b) 0 1 0

000

0 10 (c) 0 0 1

000

ri o 3

(d) I [0 I 2 4

00 (f) 0 0

00

12030

00110 (e)

0000 I

00000

r1 --7 5 51

(g)

L

3 2j

fl 2 0 2.(a)I0 1 0

Lo 0 0

I 00 (b) 0 I 0

020

I 34 (c) 00 I

000

1 (d) 0

0

5 --3 II 00

1 23 (e) 0 0 0

001

12345 1 071 3 0000 1 00000

ri --2

(g)[0 0

0I 1 --2

In Exercises .3--3, suppose that the augmented matrix for a lin ear system has been reduced by row operations to the given row echelon form. Solve the system.

I --3 4 7 3.(a) 0 1 2 2

00 I 5

1 0 8--5 6 (b)0 I 4--9 3

001 1 2

I

0 (c)

0

o

7 --2 0I 00 00

0 --8 --3 1 65 I39 000

I --3 7 (d) 0 1 4

000

r

1.2 Gaussian Elimination 23

I 0 0 --3 4.(a) 0 1 0 0

o0I 7

I 0 0 --7 8 (b) 0 I 0 3 2

0 0 I I --5

I --6 0 0 3 2

c) 0

0

0 0

1 0

0 I

4 5

7 8

000 00 0

I --3 0 0 (d) 0 0 1 0

00

In Exercises 5--8, solve the linear system by Gaussian climi nation.

5. 11+ 12+233= 8 --11--213+313= I

311--712+413=10

6. 211+233+213= 0

--2x+5xz+2x= 811+ 12+413=--I

7. x-- y+2z-- w=--I

2x+ y--2z--2w--Z

--x+2y--4z+ W= I

3x

--3w=--3

8. -- 2b + 3c = I 3a + fib -- 3c = --2 6a + 6b + 3c = 5

In Exercises 9--12, solve the linear system by Gauss--Jordan elimination.

9. ExerciseS

10. Exercise 6

l7.3xl+x+x3+x4O 511--x,+13--14=0

18.

v+3w--2x=O

2u+ v--4w+3x=O

2u+3v+2w-- 1=0

--4u--3v+5iv--4x=0

19.

2x+2v+4z0

IL'

-- -y--32=O

2w+3x+ y+ z=0

--2w+ x+3y--2:=O

20. 11+312

?14=0

x+4x3?233

=0

-- 2x -- It3 -- 13 = 0

211--412+ 13+14=0

i --212-- 13+330

21. 21 12 + 313 + 414 = 9 --

II

--213+714=11

31--31+ 13+514 8

21+ 12+413+414= 10

22.

Z3+ Z4+Z5=0

--Zi -- Z3+2Z3--3Z4+Z5=O

z+ Z2--2Z3

2Z1+2Z3-- Z3

In each parc of Exercises 23--24, the aunented matrix for a

linear system is given in which the asterisk represents an unspec

ified real number. Determine whether the system is consistent,

and if so whether the solution is unique. Answer "inconclusive" if

there is not enough information to make a decision. -1

[1 *

*

23. (a)

*

[I * *

(b)

i

II. Exercise 7

12. Exercise 8

In Exercises 13-14, determine whether the homogeneous system has nontrivial solutions by inspection (without pencil and

paper).

13.1v1--3x2+4x3-- 14=0

711+ 13--813+914=0

211+812+ 1j

t40

14.11+313--13=0

=0

In Exercises 15--22, solve the given linear system by any method,

l5.2x+ 12+313=0

11+212

=0

11=0

16. lv-- y--3z=O --x+2y--3cO 1- y-r4Z=O

P * * *1

(c) 0 I * *

L0 0 0 iJ

Il * * *

(d) 0 0 * 0

0I *

*1 [I * * H 24.01)10 1 *

0 I Ii

Fl 0 0 01

*J (c)Il 0011 [i * *

El 0 0 * (b)1* 1 0 *

L* * I *

Fl * * *

(dHiOO I

Li 0 0 1

In Exercises 25 26. determine Ihe values of a for which the system has no solutions, exactly one solution, or infinitely many

solutions

25. x+2y--

3z= 4

3x-- y+

5z= 2

4x+ y-'-(a2--14)z=a+2

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