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Vector Equation - Linear Algebra - Quiz Solution, Exercises of Linear Algebra

This is the Quiz Solution of Linear Algebra which includes Zero Vector, Linearly Dependent, Statement, Vector, Linear Combination, Expressed, Trivial Solution, Inspection, Dependent, Theorem etc. Key important points are: Vector Equation, Corresponding, Matrix Equation, Solution, Equation, Find One, Span, Columns, Bracket, Replaced

Typology: Exercises

2012/2013

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MATH 205A,B - LINEAR ALGEBRA
WINTER 2013
QUIZ 2
NAME: Section:(Circle one) A(1 : 10) B(2 : 40)
Show ALL your work CAREFULLY.
Let
A=ļ£®
ļ£°
004
0āˆ’3āˆ’1
āˆ’2 8 āˆ’5
ļ£¹
ļ£»and ~
b=ļ£®
ļ£°
4
āˆ’7
13
ļ£¹
ļ£».
(a) Write the vector equation corresponding to the matrix equation A~x =~
b.
Since the matrix Ahas 3rows, the solutions of the corresponding system have 3
variables, x1, x2and x3. The following is the corresponding vector equation:
x1ļ£®
ļ£°
0
0
āˆ’2
ļ£¹
ļ£»+x2ļ£®
ļ£°
0
āˆ’3
8
ļ£¹
ļ£»+x3ļ£®
ļ£°
4
āˆ’1
āˆ’5
ļ£¹
ļ£»=ļ£®
ļ£°
4
āˆ’7
13
ļ£¹
ļ£».
(b) Does the matrix equation A~x =~
bhave a solution? If yes, find one.
The corresponding augmented matrix is
ļ£®
ļ£°
0044
0āˆ’3āˆ’1āˆ’7
āˆ’2 8 āˆ’5 13
ļ£¹
ļ£».
Using elementary row operations, the reduced row echelon form of Acan be obtained
as follows.
ļ£®
ļ£°
0044
0āˆ’3āˆ’1āˆ’7
āˆ’2 8 āˆ’5 13
ļ£¹
ļ£»āˆ¼ļ£®
ļ£°
āˆ’2 8 āˆ’5 13
0āˆ’3āˆ’1āˆ’7
0044
ļ£¹
ļ£»āˆ¼ļ£®
ļ£°
āˆ’2 8 āˆ’5 13
0āˆ’3āˆ’1āˆ’7
0011
ļ£¹
ļ£»āˆ¼ļ£®
ļ£°
āˆ’2 8 āˆ’5 13
0āˆ’3 0 āˆ’6
0011
ļ£¹
ļ£»
āˆ¼ļ£®
ļ£°
āˆ’2 8 āˆ’5 13
0 1 0 2
0 0 1 1
ļ£¹
ļ£»āˆ¼ļ£®
ļ£°
āˆ’2 8 0 18
0 1 0 2
0 0 1 1
ļ£¹
ļ£»āˆ¼ļ£®
ļ£°
āˆ’2 0 0 2
0 1 0 2
0 0 1 1
ļ£¹
ļ£»āˆ¼ļ£®
ļ£°
1 0 0 āˆ’1
0 1 0 2
0 0 1 1
ļ£¹
ļ£».
THE solution is x1=āˆ’1, x2= 2 and x3= 1.
(c) Do the columns of Aspan R3? Explain.
YES, they do. From part (b), the reduced row echelon form of Ais
ļ£®
ļ£°
1 0 0
0 1 0
0 0 1
ļ£¹
ļ£».
Date: January 18, 2013.
1
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MATH 205A,B - LINEAR ALGEBRA

WINTER 2013

QUIZ 2

NAME: Section:(Circle one) A(1 : 10) B(2 : 40)

Show ALL your work CAREFULLY.

Let

A =

ļ£» (^) and ~b =

(a) Write the vector equation corresponding to the matrix equation A~x = ~b.

Since the matrix A has 3 rows, the solutions of the corresponding system have 3 variables, x 1 , x 2 and x 3. The following is the corresponding vector equation:

x 1

ļ£» (^) + x 2

ļ£» (^) + x 3

(b) Does the matrix equation A~x = ~b have a solution? If yes, find one.

The corresponding augmented matrix is ļ£®

ļ£°

Using elementary row operations, the reduced row echelon form of A can be obtained as follows. ļ£®

ļ£°

THE solution is x 1 = āˆ’ 1 , x 2 = 2 and x 3 = 1.

(c) Do the columns of A span R^3? Explain.

YES, they do. From part (b), the reduced row echelon form of A is ļ£® ļ£°

Date: January 18, 2013. 1

It follows that for any arbitrary vector ~b, the equation A~x = ~b always has a (unique) solution. Thus, every ~b is a linear combination of the columns of A.