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Title: Orthonormal bases and orthogonal matrices
Description: Linear algebra course
Description: Linear algebra course
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�
about the axis determined by
ih =
(a) Verify that
=
is orthogonal to
x
and calculate the compo
Define
-�[ ]
§2§2
...
f;}
...
Write the change of coordinates matrix P for
the change from :B to the standard basis S in
the form P=
�[
·
]
...
(c) Write the :B-matrix of
L, [L]2J
...
(d) Determine the standard matrix of
[L]s
...
/3
1/
...
Determine another
3
orthonormal basis for IR
...
/6
1/
...
Homework Problems
gonal
...
(a)
(b)
( c)
{[�]·[-i]}
(c ) y =
·
{[-:J
...
(a) A=
]
u
·
{ [ �l n l }
}o
},
(b) A=
Find the
(c ) A=
coordinates of each of the following vectors with
m
Ul
respect to the orthonormal basis '13
...
j
-
3
orthogonal, indicate how the columns of A fail to
, 1 , -2
1 0
·
-2
2
A is orthogonal by calculating AT A
...
(d) A=
(e) A=
[2/1;-VSYS -1/
- 1-VS
2/YS -1/YS]
[-11-VS -21-VS
1/2 1/2 ]
[-1/2 1/2
[
[
]
1/
...
/6
-1/
...
1/
...
/6
1/
...
/3 1/
...
/3 1/
...
/3 -1/
...
(b) Let v;
=
Ul
...
w;/llw;ll so that '13 = {v1, 1!2, 1/3} is an
orthonormal basis
...
D2 (a) Prove that if P is an orthogonal matrix, then
detP= ± 1
...
y =Py to show that if an
n x n matrix Pis orthogonal, then llPxll = 11111
for every 1E JRn
...
D4 Prove that an n x n matrix P is orthogonal if and
only if the rows of Pform an orthonormal set
Title: Orthonormal bases and orthogonal matrices
Description: Linear algebra course
Description: Linear algebra course