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Found in: Page 93

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

In Exercises 27 and 28, view vectors in $${\mathbb{R}^n}$$ as $$n \times 1$$ matrices. For $${\mathop{\rm u}\nolimits}$$ and $${\mathop{\rm v}\nolimits}$$ in $${\mathbb{R}^n}$$, the matrix product $${{\mathop{\rm u}\nolimits} ^T}v$$ is a $$1 \times 1$$ matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product $${{\mathop{\rm uv}\nolimits} ^T}$$ is a $$n \times n$$ matrix, called the outer product of u and v. The products $${{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits}$$ and $${{\mathop{\rm uv}\nolimits} ^T}$$ will appear later in the text.28. If u and v are in $${\mathbb{R}^n}$$, how are $${{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits}$$ and $${{\mathop{\rm v}\nolimits} ^T}{\mathop{\rm u}\nolimits}$$ related? How are $${{\mathop{\rm uv}\nolimits} ^T}$$ and $${\mathop{\rm v}\nolimits} {{\mathop{\rm u}\nolimits} ^T}$$ related?

The relation of inner and outer product is $${{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} = {{\mathop{\rm v}\nolimits} ^T}{\mathop{\rm u}\nolimits}$$ and $${\mathop{\rm u}\nolimits} {{\mathop{\rm v}\nolimits} ^T} = {\mathop{\rm v}\nolimits} {{\mathop{\rm u}\nolimits} ^T}$$, respectively.

See the step by step solution

Step 1: Determine the relation of the inner product

Theorem 3 states that $$A$$ and $$B$$ denotes matrices whose sizes are appropriate for the following sums and products.

1. $${\left( {{A^T}} \right)^{^T}} = A$$.
2. $${\left( {A + B} \right)^T} = {A^T} + {B^T}$$.
3. For any scalar $$r$$, $${\left( {rA} \right)^T} = r{A^T}$$.
4. $${\left( {AB} \right)^T} = {B^T}{A^T}$$.

The inner product $${{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits}$$ is a real number since it equals its transpose.

Use theorem 3 to obtain the relation of the inner product as follows:

\begin{aligned}{c}{{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} = {\left( {{{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} } \right)^T}\\ = {{\mathop{\rm v}\nolimits} ^T}{\left( {{{\mathop{\rm u}\nolimits} ^T}} \right)^T}\\ = {{\mathop{\rm v}\nolimits} ^T}{\mathop{\rm u}\nolimits} \end{aligned}

Step 2: Determine the relation of the outer product

A $$n \times n$$ matrix is an outer product $${\mathop{\rm u}\nolimits} {{\mathop{\rm v}\nolimits} ^T}$$.

Use theorem 3 to obtain the relation of the outer product as follows:

\begin{aligned}{c}{\mathop{\rm u}\nolimits} {{\mathop{\rm v}\nolimits} ^T} = {\left( {{\mathop{\rm u}\nolimits} {{\mathop{\rm v}\nolimits} ^T}} \right)^T}\\ = {\left( {{{\mathop{\rm v}\nolimits} ^T}} \right)^T}{{\mathop{\rm u}\nolimits} ^T}\\ = {\mathop{\rm v}\nolimits} {{\mathop{\rm u}\nolimits} ^T}\end{aligned}

Thus, the relation of inner and outer products is $${{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} = {{\mathop{\rm v}\nolimits} ^T}{\mathop{\rm u}\nolimits}$$ and $${\mathop{\rm u}\nolimits} {{\mathop{\rm v}\nolimits} ^T} = {\mathop{\rm v}\nolimits} {{\mathop{\rm u}\nolimits} ^T}$$, respectively.