Question

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 \({\mathbb{R}^n}\).

Short answer

1. The given statement is true.
2. The given statement is true.
3. The given statement is true.
4. The given statement is false.
5. The given statement is false.

Step-by-step solution

Check whether the statement is true or false

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.

Check whether the statement is true or false

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.

Check whether the statement is true or false

c)

By Theorem 8, there is unique orthogonal decomposition.

Thus, the given statement (c) is true.

Check whether the statement is true or false

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.

Check whether the statement is true or 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.