Homework #3a - Describing Solution Sets

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Section 1.5 – Solutions of Linear Equations

1.

Determine if the following systems has a non-trivial solution: i.

2

x

1

2

x

1

x

1

2

4

x

2

3

x x

2

2

3

9

x

3

4

x x

3

3

0

0

0 ii.

2

4

x

1

x

1

2

x

4

3

x

2

x

2

2

4

3

8

x x

3

x

3

3

0

0

0

2.

Describe all solutions of

A

x

0 in parametric vector form, where

A

is row equivalent to the given matrix. Careful, the matrix below is NOT the augmented matrix, but the coefficient matrix! i.

1

0

3

1

8

2

5

4

1

ii.

1

0

0

0

4

0

0

0

2

1

0

0

0

0

0

0

3

0

1

0

0

5

1

4

3.

Describe and compare the solution sets of

x

1

5

x

2

3

x

3

0 and

x

1

5

x

2

3

x

3

 

2 .

4.

Describe the solutions of the following system in parametric vector form, and compare it to that of exercise 13.ii:

2

x

1

4

x

1

2

x

2

4

x

2

3

x

2

4

x

3

8

x

3

3

x

3

8

16

12

.

2

5.

Find the parametric equation of the line through

a

parallel to

b

: i.

a



0

2

,

b



3

5

 ii.

a

3

2

,

b



6

7

6.

Does the equation

A

x

0 have a non-trivial solution, and does the equation solution for every possible

b

? i.

A

is a 3

3 matrix with 3 pivot positions

A x

b

have at least one ii.

A

is a

4

4

matrix with 3 pivot positions iii.

A

is a 2

5 matrix with 2 pivot positions

3

7.

Mark each statement as

True

or

False

and justify your answer. i.

ii.

An example of a linear combination of vectors

v

1

and

v

2

The weights

c

1

,

c

2

, …

c p

in a linear combination

c

1

v

1

is the vector

1

2

v

1

.

...

c p

v p

cannot all be zero. iii.

iv.

v.

vi.

vii.

The equation

x

p

t v

describes a line through

v

parallel to

p

.

The equation

A x

b

is referred to as a vector equation.

If

A

is a

m

n

matrix and if the equation

A x

b

 

is inconsistent for some

b

in

R m

, then

A

cannot have a pivot position in every row.

The solution set of a linear system whose augmented matrix is the solution set of

A x

b

if

A

a

1

a

2

a

3

.

a

1

a

2

a

3

b

, is the same as

If the equation

A x

b

is consistent, then

b

is in the set spanned by the columns of

A

.

4

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