Question

In Exercises 19 and 20, all vectors are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

1. \(v \cdot {\rm{v}} = {\left\| {\rm{v}} \right\|^2}\).
2. For any scalar \(c\),\({\rm{u}} \cdot \left( {c{\rm{v}}} \right) = c\left( {{\rm{u}} \cdot {\rm{v}}} \right)\).
3. If the distance from \({\rm{u}}\) to \({\rm{v}}\) equals the distance from \({\rm{u}}\) to \( - {\rm{v}}\), then \({\rm{u}}\) and \({\rm{v}}\) are orthogonal.
4. For a square matrix \(A\), vectors in \({\rm{Col }}A\) are orthogonal to vectors in Nul \(A\).
5. If vectors \({{\rm{v}}_1}..........,{{\rm{v}}_p}\) span a subspace \(W\) and if \({\rm{x}}\) is orthogonal to each \({{\rm{v}}_j}\) for \(j = 1,..........,p\), then \({\rm{x}}\) is in \({W^ \bot }\)?.

Short answer

1. True, by using the definition of the length of the vector.
2. True, by using Theorem 1(c).
3. True, by using the definition of orthogonal vectors.
4. False, as the case fails when the square matrix is not symmetric.
5. True, by using the definition for the spans of any vector.

Step-by-step solution

Definition of Orthogonal Set

The two vectors \({\rm{u and v}}\) are Orthogonal if:

\(\begin{aligned}{l}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} = {\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2}\\{\rm{and}}\\{\bf{u}} \cdot {\bf{v}} = 0\end{aligned}\).

 Verification of statement (a)

The definition for the length of any vector states that:

\({\left\| {\bf{v}} \right\|^2} = {\bf{v}} \cdot {\bf{v}}\)

Hence, the given statement is true.

 Verification of statement (b)

The Theorem 1(c)resembles that:

\(\left( {c{\bf{u}}} \right) \cdot {\bf{v}} = c\left( {{\bf{u}} \cdot {\bf{v}}} \right) = {\bf{u}} \cdot \left( {c{\bf{v}}} \right)\)

Hence, the given statement is true.

 Verification of statement (c)

The definition of the Orthogonal Vectorsstates that:

\(\begin{aligned}{l}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} = {\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2}\\{\rm{and}}\\{\bf{u}} \cdot {\bf{v}} = 0\end{aligned}\)

Hence, the given statement is true.

 Verification of statement (d)

The given statement is only valid for the particular type of square matrices. Just in case if there is a matrix of type:

\(A = \left( {\begin{aligned}{*{20}{c}}1&1\\0&0\end{aligned}} \right)\)

The given statement would not be true.

Hence, the given statement is False.

 Verification of statement (e)

The definition for the spans of any vector states that for the given condition in question, the span:

\({W^ \bot } \subseteq {\mathbb{R}^n}\)

Hence, the given statement is true.