Guided Proof Let u, v be the Euclidean innerproduct on Rn. Use the fact that u, v = uTv

Chapter 5, Problem 95

(choose chapter or problem)

Guided Proof Let \(\langle\mathbf{u}, \mathbf{v}\rangle\) be the Euclidean inner product on \(R^{n}\). Use the fact that \(\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u}^{T} \mathbf{v}\) to prove that for any n X n matrix A,

(a) \(\left\langle A^{T} A \mathbf{u}, \mathbf{v}\right\rangle=\langle\mathbf{u}, A \mathbf{v}\rangle\)

and

(b) \(\left\langle A^{T} A \mathbf{u}, \mathbf{u}\right\rangle=\|A \mathbf{u}\|^{2}\).

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 \(\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u}^{T} \mathbf{v}\) and Property 4 of Theorem 2.6.

(ii) To prove part (b), make use of the property \(\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u}^{T} \mathbf{v}\), Property 4 of Theorem 2.6, and Property 4 of Theorem 5.3.

Text Transcription:

langle u, v rangle

R^n

langle u, v rangle = u^{T} v

langle A^{T} Au, v rangle = langle u, Av rangle

langle A^{T} Au, u rangle = ||Au||^2

langle u, v rangle = u^{T} v