Exercises 45 − 49 are based on the optional material. Let A = [ A 1 , A 2 ] be a ( 2 × 2 ) matrix such that A T A = I , and define T : R 2 → R 2 by T ( x ) = A x . a) Show that { A 1 , A 2 } is an orthonormal set. b) Use Theorem 16 to show that T is an orthogonal transformation. Theorem 16 Let T : R 2 → R 2 be a linear transformation. Then T is orthogonal if and only if ‖ T ( e 1 ) ‖ = ‖ T ( e 1 ) ‖ = 1 and T ( e 1 ) is perpendicular to T ( e 2 ) .
Exercises 45 − 49 are based on the optional material. Let A = [ A 1 , A 2 ] be a ( 2 × 2 ) matrix such that A T A = I , and define T : R 2 → R 2 by T ( x ) = A x . a) Show that { A 1 , A 2 } is an orthonormal set. b) Use Theorem 16 to show that T is an orthogonal transformation. Theorem 16 Let T : R 2 → R 2 be a linear transformation. Then T is orthogonal if and only if ‖ T ( e 1 ) ‖ = ‖ T ( e 1 ) ‖ = 1 and T ( e 1 ) is perpendicular to T ( e 2 ) .
Solution Summary: The author explains that the set leftA_1, A2right is an orthonormal set.
Exercises
45
−
49
are based on the optional material.
Let
A
=
[
A
1
,
A
2
]
be a
(
2
×
2
)
matrix such that
A
T
A
=
I
, and define
T
:
R
2
→
R
2
by
T
(
x
)
=
A
x
.
a) Show that
{
A
1
,
A
2
}
is an orthonormal set.
b) Use Theorem
16
to show that
T
is an orthogonal transformation.
Theorem
16
Let
T
:
R
2
→
R
2
be a linear transformation. Then
T
is orthogonal if and only if
‖
T
(
e
1
)
‖
=
‖
T
(
e
1
)
‖
=
1
and
T
(
e
1
)
is perpendicular to
T
(
e
2
)
.
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Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY