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.
An example of a linear combination of vectors
 v
1
and v

2 ii.
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.
The equation
 x

 p
 t v

describes a line through v
 
parallel to p . iv.
The equation

A x
 b

is referred to as a vector equation. v.
If A is a m
 n matrix and if the equation

A x
 b
 
is inconsistent for some b in m
R , then A cannot have a pivot position in every row. vi.
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 vii.
If the equation

A x

 b

is consistent, then b is in the set spanned by the columns of A .
4