22. Question: In Exercises 21 and 22, all vectors are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

1. If W is a subspace of \({\mathbb{R}^n}\) and if v is in both W and \({W^ \bot }\), then v must be the zero vector.
2. In the Orthogonal Decomposition Theorem, each term in formula (2) for \(\widehat {\mathop{\rm y}\nolimits} \) is itself an orthogonal projection of y onto a subspace of \(W\).
3. If \({\mathop{\rm y}\nolimits} = {{\bf{z}}_1} + {{\bf{z}}_2}\), where \({{\bf{z}}_1}\) is in a subspace W and \({{\bf{z}}_2}\) is in \({W^ \bot }\), then \({{\bf{z}}_1}\) must be the orthogonal projection of \({\mathop{\rm y}\nolimits} \) onto W.
4. The best approximation to y by elements of a subspace W is given by the vector \({\mathop{\rm y}\nolimits} - {{\mathop{\rm proj}\nolimits} _W}{\mathop{\rm y}\nolimits} \).
5. If an \(n \times p\) matrix \(U\) has orthonormal columns, then \(U{U^T}{\mathop{\rm x}\nolimits} = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\).

a)

The equality \(\widehat {\bf{y}} - {\widehat {\bf{y}}_1} = {{\bf{z}}_1} - {\bf{z}}\) demonstrates that the vector \({\mathop{\rm v}\nolimits} = \widehat {\bf{y}} - {\widehat {\bf{y}}_1}\) is in W and is in \({W^ \bot }\). Thus, \({\bf{v}} = 0\).

Thus, the given statement (a) is true.

b)

If \(\dim > 1\), then every term in \(\widehat {\bf{y}}\) is itself an orthogonal projection of y onto a subspace spanned by several of the u’s in the basis for \(W\).

Thus, the given statement (b) is true.

c)

By Theorem 8, there is unique orthogonal decomposition.

Thus, the given statement (c) is true.

d)

The best approximation theorem states that the \({{\mathop{\rm proj}\nolimits} _w}{\bf{y}}\) is the best approximation to y.

Thus, the given statement (d) is false.

e)

This statement holds only when the column space of \(U\) is in \({\bf{x}}\). When \(n > p\)then, the column space of \(U\) would not be all of \({\mathbb{R}^n}\). Therefore, the statement would not be true for all \({\bf{x}}\) in \({\mathbb{R}^n}\).

Thus, the given statement (e) is false.