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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