Consider the vector space ℝ n with inner product 〈 x , y 〉 = x T y . Show that for any n × n matrix A , (a) 〈 A x , y 〉 = 〈 x , A T y 〉 (b) 〈 A T A x , x 〉 = ‖ A x ‖ 2
Consider the vector space ℝ n with inner product 〈 x , y 〉 = x T y . Show that for any n × n matrix A , (a) 〈 A x , y 〉 = 〈 x , A T y 〉 (b) 〈 A T A x , x 〉 = ‖ A x ‖ 2
Solution Summary: The author analyzes the vector space Rn with inner product langle x,yrangle =xT
Consider the vector space
ℝ
n
with inner product
〈
x
,
y
〉
=
x
T
y
.
Show that for any
n
×
n
matrix A, (a)
〈
A
x
,
y
〉
=
〈
x
,
A
T
y
〉
(b)
〈
A
T
A
x
,
x
〉
=
‖
A
x
‖
2
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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