Question

Use the transpose definition of the inner product to verify parts (b) and (c) of Theorem 1. Mention the appropriate facts from Chapter 2.

Short answer

Using the definition of inner product, verified that:

1. \(\left( {{\bf{u}} + {\bf{v}}} \right) \cdot {\bf{w}} = {\bf{u}} \cdot {\bf{w}} + {\bf{v}} \cdot {\bf{w}}\)
2. \(\left( {c{\bf{u}}} \right) \cdot {\bf{v}} = c\left( {{\bf{u}} \cdot {\bf{v}}} \right)\)

Step-by-step solution

 Verification of theorem 1(b).

Part (b) of theorem 1 is,\(\left( {{\bf{u}} + {\bf{v}}} \right) \cdot {\bf{w}} = {\bf{u}} \cdot {\bf{w}} + {\bf{v}} \cdot {\bf{w}}\).

Prove it by using the definition of Inner Product.

\(\begin{aligned}{c}\left( {{\bf{u}} + {\bf{v}}} \right) \cdot {\bf{w}} = {\left( {{\bf{u}} + {\bf{v}}} \right)^T} \cdot {\bf{w}}\\ = \left( {{{\bf{u}}^T} + {{\bf{v}}^T}} \right) \cdot {\bf{w}}\\ = {{\bf{u}}^T} \cdot {\bf{w}} + {{\bf{v}}^T} \cdot {\bf{w}}\\ = {\bf{u}} \cdot {\bf{w}} + {\bf{v}} \cdot {\bf{w}}\end{aligned}\)

Hence, part (b) of theorem 1 has been verified.

 Verification of theorem 1(c).

Part (c) of theorem 1 is,\(\left( {c{\bf{u}}} \right) \cdot {\bf{v}} = c\left( {{\bf{u}} \cdot {\bf{v}}} \right)\).

Prove it by using the definition of Inner Product.

\(\begin{aligned}{c}\left( {c{\bf{u}}} \right) \cdot {\bf{v}} = {\left( {c{\bf{u}}} \right)^T} \cdot {\bf{v}}\\ = c\left( {{{\bf{u}}^T}{\bf{v}}} \right)\\ = c\left( {{\bf{u}} \cdot {\bf{v}}} \right)\end{aligned}\)

Hence, part (c) of theorem 1 has been verified.