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Linear Algebra With Applications
Found in: Page 22
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

a. Find all solutionsx1,x2,x3,x4 of the system .

x2=12(x1+x3),x3=12(x2+x4)

b. In partrole="math" localid="1659677484607" (a) , is there a solution with x1=1andx4=13 ?

a. For arbitrary x3 and x4 , x1=3x3−2x4,x2=2x3−x4

b. Yes, x1=1,x2=5,x3=9,x4=13

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

(a)Step 1: Consider the equations to find other equations

Find the expression for, x1.

x2=12(x1+x3)2x2=x1+x3x1=2x2−x3       ......(1)

Find the expression for, x2 .

x3=12(x2+x4)2x3=x2+x4x2=2x3−x4           ......(2)

Step 2: Express the variable in terms of other variables

Re-write x1in terms of x3andx4 .

x1=2x2−x3x1=2(2x3−x4)−x3=3x3−2x4       ......(3)

Therefore, x1=3x3−2x4,x2=2x3−x4

(b)Step 3: Using the above equations, find the values of the variables

Consider the given values of the variables.

x1=1 and x4=13

Substitute these values in the above equations to find the values of the remaining variables.

Consider the equation (3).

x1=3x3−2x41=3x3−2×13x3=273=9

Consider the equation (2).

x2=2x3−x4=2×9−13=18−13=5

Therefore, x1=1,x2=5,x3=9,x4=13.

Most popular questions for Math Textbooks

In exercises, 1 through 10, findall solutions of the linear systems using elimination.T hen check your solutions.

10.|x+2y+3z=12x+4y+7z=23x+7y+11z=8|

Determine whether the statements that follow are true or false, and justify your answer.

16: There exists a 2×2matrix such that

A[11]=[12] and A[22]=[21]

Consider a solutionx1→of the linear systemAx→=b→. Justify the facts stated in parts (a) and (b):

a. Ifx→his a solution of the systemAx→=0→, thenx1→+xh→ is a solution of the system A=x→=b→.

b. Ifx2→is another solution of the systemAx→=b→, thenx1→+xh→is a solution of the system Ax→+0→.

c. Now suppose A is a2×2matrix. A solution vectorx1→of the systemAx→+b→is shown in the accompanying figure. We are told that the solutions of the systemAx→=0→form the line shown in the sketch. Draw the line consisting of all solutions of the systemAx→=b→.

If you are puzzled by the generality of this problem, think about an example first:

A=(1    23    6),b→=[39]andx1→=[11]

Determine whether the statements that follow are true or false, and justify your answer.

17: Rank |222222222|=2

Use the concept of a continuous dynamical system.Solve the differential equation dxdt=−kx . Solvethe system dx→dt=Ax→ when A is diagonalizable over R,and sketch the phase portrait for 2 × 2 matrices A.

Solve the initial value problems posed in Exercises 1through 5. Graph the solution.

4 . dydt=0.8t with y(0)=-0.8
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