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?

Theorem 3states 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:

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

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.