Consider a consistent system A x → = b → . a. Show that this system has a solution x → 0 in ( ker A ) ⊥ .Hint: An arbitrary solution x → of the system can bewritten as x → = x → h + x → 0 . where x → h is in ker(A) and x → 0 isin ( ker A ) ⊥ . b. Show that the system A x → = b → has only one solutionin ( ker A ) ⊥ . Hint: If x → 0 and x → 1 are two solutions in ( ker A ) ⊥ , think about x → 1 − x → 0 . c. If x → 0 is the solution in ( ker A ) ⊥ and x → 1 is anothersolution of the system A x → = b → . show that ‖ x → 0 ‖ < ‖ x → 1 ‖ . The vector x → 0 called the minimal solution ofthe linear system A x → = b → .
Consider a consistent system A x → = b → . a. Show that this system has a solution x → 0 in ( ker A ) ⊥ .Hint: An arbitrary solution x → of the system can bewritten as x → = x → h + x → 0 . where x → h is in ker(A) and x → 0 isin ( ker A ) ⊥ . b. Show that the system A x → = b → has only one solutionin ( ker A ) ⊥ . Hint: If x → 0 and x → 1 are two solutions in ( ker A ) ⊥ , think about x → 1 − x → 0 . c. If x → 0 is the solution in ( ker A ) ⊥ and x → 1 is anothersolution of the system A x → = b → . show that ‖ x → 0 ‖ < ‖ x → 1 ‖ . The vector x → 0 called the minimal solution ofthe linear system A x → = b → .
Solution Summary: The author explains that the consistent system Astackrelto x=
Consider a consistent system
A
x
→
=
b
→
. a. Show that this system has a solution
x
→
0
in
(
ker
A
)
⊥
.Hint: An arbitrary solution
x
→
of the system can bewritten as
x
→
=
x
→
h
+
x
→
0
. where
x
→
h
is in ker(A) and
x
→
0
isin
(
ker
A
)
⊥
. b. Show that the system
A
x
→
=
b
→
has only one solutionin
(
ker
A
)
⊥
. Hint: If
x
→
0
and
x
→
1
are two solutions in
(
ker
A
)
⊥
, think about
x
→
1
−
x
→
0
. c. If
x
→
0
is the solution in
(
ker
A
)
⊥
and
x
→
1
is anothersolution of the system
A
x
→
=
b
→
. show that
‖
x
→
0
‖
<
‖
x
→
1
‖
. The vector
x
→
0
called the minimal solution ofthe linear system
A
x
→
=
b
→
.
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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