Consider the linear system A x → = b → , where A = [ 1 3 2 6 ] and b → = [ 10 20 ] a. Draw a sketch showing the following subsets of ℝ 2 : • the kernelof A, and ( ker A ) ⊥ •theimage of A T • the solution set S of the system A x → = b → b. What relationship do you observe between ker(A) and i m ( A T ) ? Explain. c. What relationship do you observe between ker( A)and S? Explain. d. Find the unique vector x → 0 in the intersection of S and ( ker A ) ⊥ . Show x → 0 on your sketch. e. What can you say about the length of x → 0 comparedwith the length of all other vectors in S?
Consider the linear system A x → = b → , where A = [ 1 3 2 6 ] and b → = [ 10 20 ] a. Draw a sketch showing the following subsets of ℝ 2 : • the kernelof A, and ( ker A ) ⊥ •theimage of A T • the solution set S of the system A x → = b → b. What relationship do you observe between ker(A) and i m ( A T ) ? Explain. c. What relationship do you observe between ker( A)and S? Explain. d. Find the unique vector x → 0 in the intersection of S and ( ker A ) ⊥ . Show x → 0 on your sketch. e. What can you say about the length of x → 0 comparedwith the length of all other vectors in S?
Solution Summary: The author denotes the kernel of T by ker(A) and it is the solution set of the linear system.
Consider the linear system
A
x
→
=
b
→
, where
A
=
[
1
3
2
6
]
and
b
→
=
[
10
20
]
a. Draw a sketch showing the following subsets of
ℝ
2
: • the kernelof A, and
(
ker
A
)
⊥
•theimage of
A
T
• the solution set S of the system
A
x
→
=
b
→
b. What relationship do you observe between ker(A) and
i
m
(
A
T
)
? Explain. c. What relationship do you observe between ker( A)and S? Explain. d. Find the unique vector
x
→
0
in the intersection of S and
(
ker
A
)
⊥
. Show
x
→
0
on your sketch. e. What can you say about the length of
x
→
0
comparedwith the length of all other vectors in S?
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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