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

1. If z is orthogonal to \({{\mathop{\rm u}\nolimits} _1}\) and to \({{\mathop{\rm u}\nolimits} _2}\) and if \(W = {\mathop{\rm Span}\nolimits} \left\{ {{{\mathop{\rm u}\nolimits} _1},{{\mathop{\rm u}\nolimits} _2}} \right\}\), then z must be in \({W^ \bot }\).
2. For each y and each subspace \(W\), the vector \({\mathop{\rm y}\nolimits} - {{\mathop{\rm proj}\nolimits} _W}{\mathop{\rm y}\nolimits} \) is orthogonal to \(W\).
3. The orthogonal projection \(\widehat {\mathop{\rm y}\nolimits} \) of \({\mathop{\rm y}\nolimits} \) onto a subspace W can sometimes depend on the orthogonal basis for W used to compute \(\widehat {\mathop{\rm y}\nolimits} \).
4. If y is in a subspace W, then the orthogonal projection of y onto W is y itself.
5. If the columns of an \(n \times p\) matrix \(U\) are orthonormal, then \(U{U^T}{\mathop{\rm y}\nolimits} \) is the orthogonal projection of y onto the column space of \(U\).

a)

The vector \({\bf{x}}\) is in \({W^ \bot }\), such that if \({\bf{x}}\) is orthogonal to any vector in the set that spans \(W\).

Also, \({\mathbb{R}^n}\) has a subspace \({W^ \bot }\).

Thus, the given statement (a) is true.

b)

The Orthogonal Decomposition theoremstates that, suppose that \(W\) is a subspace of \({\mathbb{R}^n}\). Then each \({\bf{y}}\) in \({\mathbb{R}^n}\) can be expressed uniquely in the form:

\({\bf{y}} = \widehat {\bf{y}} + {\bf{z}}\) … (1)

With \(\widehat {\bf{y}}\) is in \(W\) and \({\bf{z}}\) is in \({W^ \bot }\). In particular, when \(\left\{ {{{\mathop{\rm u}\nolimits} _1}, \ldots ,{{\mathop{\rm u}\nolimits} _p}} \right\}\) is anorthogonal basis of \(W\), then;

\(\widehat {\bf{y}} = \frac{{{\mathop{\rm y}\nolimits} \cdot {{\mathop{\rm u}\nolimits} _1}}}{{{{\mathop{\rm u}\nolimits} _1} \cdot {{\mathop{\rm u}\nolimits} _1}}}{{\mathop{\rm u}\nolimits} _1} + \ldots + \frac{{{\mathop{\rm y}\nolimits} \cdot {{\mathop{\rm u}\nolimits} _p}}}{{{{\mathop{\rm u}\nolimits} _p} \cdot {{\mathop{\rm u}\nolimits} _p}}}{{\mathop{\rm u}\nolimits} _p}\) … (2)

And, \({\bf{z}} = {\bf{y}} - \widehat {\bf{y}}\).

Thus, the given statement (b) is true.

c)

It is observed from the uniqueness of the decomposition \({\mathop{\rm y}\nolimits} = \widehat {\mathop{\rm y}\nolimits} + {\mathop{\rm z}\nolimits} \) demonstrates that theorthogonal projection \(\widehat {\mathop{\rm y}\nolimits} \)depends only on \(W\) but not on the particular basis used in \(\widehat {\mathop{\rm y}\nolimits} = \frac{{{\mathop{\rm y}\nolimits} \cdot {{\mathop{\rm u}\nolimits} _1}}}{{{{\mathop{\rm u}\nolimits} _1} \cdot {{\mathop{\rm u}\nolimits} _1}}}{{\mathop{\rm u}\nolimits} _1} + \ldots + \frac{{{\mathop{\rm y}\nolimits} \cdot {{\mathop{\rm u}\nolimits} _p}}}{{{{\mathop{\rm u}\nolimits} _p} \cdot {{\mathop{\rm u}\nolimits} _p}}}{{\mathop{\rm u}\nolimits} _p}\).

Thus, the given statement (c) is false.

d)

When \({\bf{y}}\) is in \(W = {\mathop{\rm Span}\nolimits} \left\{ {{{\bf{u}}_1}, \ldots ,{{\bf{u}}_p}} \right\}\), then \({{\mathop{\rm proj}\nolimits} _W}{\bf{y}} = {\bf{y}}\).

Thus, the given statement (d) is true.

e)

Theorem 4 holds for column space \(W\) of \(U\) since the columns of \(U\) are linearly independent and therefore, constitute a basis for \(W\).

Thus, the given statement (e) is true.