23. Question: In Exercises 23 and 24, all vectors are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

1. Not every linearly independent set in \({\mathbb{R}^n}\) is an orthogonal set.
2. If y is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.
3. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.
4. A matrix with orthonormal columns is an orthogonal matrix.
5. If L is a line through 0 and if \(\widehat {\mathop{\rm y}\nolimits} \) is the orthogonal projection of y onto L, then \(\left\| {\widehat {\mathop{\rm y}\nolimits} } \right\|\) gives the distance from y to L.

a)

Consider a counterexample that \({\bf{y}} = \left( {\begin{array}{*{20}{c}}7\\6\end{array}} \right)\)and \({\bf{u}} = \left( {\begin{array}{*{20}{c}}4\\2\end{array}} \right)\). Then, \(\left\{ {{\bf{y}},{\bf{u}}} \right\}\) is linearly independent sets but not an orthogonal set because \({\bf{y}} \cdot {\bf{u}} \ne 0\).

Thus, the given statement (a) is true.

b)

Theorem 5states that consider \(\left\{ {{{\mathop{\rm u}\nolimits} _1},{{\mathop{\rm u}\nolimits} _2}, \ldots ,{{\mathop{\rm u}\nolimits} _p}} \right\}\) as anorthogonal basisfor a subspace \(W\) of \({\mathbb{R}^n}\), then theweightsin the linear combination \({\mathop{\rm y}\nolimits} = {c_1}{{\bf{u}}_1} + \cdots + {c_1}{{\bf{u}}_p}\)for every y in \(W\) is denoted by \({c_j} = \frac{{{\bf{y}} \cdot {{\bf{u}}_j}}}{{{{\bf{u}}_j} \cdot {{\bf{u}}_j}}}\left( {j = 1, \ldots ,p} \right)\).

Thus, the given statement (b) is true.

c)

If the nonzero vector in an orthogonal set isnormalized to have unit length, then the new vectors remain orthogonal.

Thus, the given statement (c) is false.

d)

Asquare invertible matrix\(U\)is an orthogonal matrix, such that \({U^{ - 1}} = {U^T}\). Such a square matrix contains orthonormal columns.

The square matrix with orthonormal columns is orthogonal.

Thus, the given statement (d) is false.

e)

It is known that the distance fromy to L is equal to the length of the perpendicular line segment from yto the orthogonal projection \(\widehat {\bf{y}}\), that is \(\left\| {{\bf{y}} - \widehat {\bf{y}}} \right\|\).

Thus, the given statement (e) is false.