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Linear Algebra and its Applications
Found in: Page 93
Linear Algebra and its Applications

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

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Short Answer

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.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

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.

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