Guided proof Let 〈 u , v 〉 be the Euclidean inner product on R n . Use the fact that 〈 u , v 〉 = u T v to prove that for any n × n matrix A , (a) 〈 A T A u , v 〉 = 〈 u , A v 〉 and (b) 〈 A T A u , u 〉 = | | A u | | 2 Getting started: To prove (a) and (b), make use of both the properties of transposes (Theorem 2.6) and the properties of the dot product (Theorem 5.3). (i) To prove part (a), make repeated use of the property 〈 u , v 〉 = u T v and Property 4 of Theorem 2.6. (ii) To prove part (b), make use of the property 〈 u , v 〉 = u T v , Property 4 of Theorem 2.6, and Property 4 of Theorem 5.3.
Guided proof Let 〈 u , v 〉 be the Euclidean inner product on R n . Use the fact that 〈 u , v 〉 = u T v to prove that for any n × n matrix A , (a) 〈 A T A u , v 〉 = 〈 u , A v 〉 and (b) 〈 A T A u , u 〉 = | | A u | | 2 Getting started: To prove (a) and (b), make use of both the properties of transposes (Theorem 2.6) and the properties of the dot product (Theorem 5.3). (i) To prove part (a), make repeated use of the property 〈 u , v 〉 = u T v and Property 4 of Theorem 2.6. (ii) To prove part (b), make use of the property 〈 u , v 〉 = u T v , Property 4 of Theorem 2.6, and Property 4 of Theorem 5.3.
Solution Summary: The author analyzes the Euclidean inner product langle u,vrangle =uTv.
Guided proof Let
〈
u
,
v
〉
be the Euclidean inner product on
R
n
. Use the fact that
〈
u
,
v
〉
=
u
T
v
to prove that for any
n
×
n
matrix
A
,
(a)
〈
A
T
A
u
,
v
〉
=
〈
u
,
A
v
〉
and
(b)
〈
A
T
A
u
,
u
〉
=
|
|
A
u
|
|
2
Getting started: To prove (a) and (b), make use of both the properties of transposes (Theorem 2.6) and the properties of the dot product (Theorem 5.3).
(i) To prove part (a), make repeated use of the property
〈
u
,
v
〉
=
u
T
v
and Property 4 of Theorem 2.6.
(ii) To prove part (b), make use of the property
〈
u
,
v
〉
=
u
T
v
, Property 4 of Theorem 2.6, and Property 4 of Theorem 5.3.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
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