Question

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 \(U\).

Short answer

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

Step-by-step solution

Check whether the statement is true or false

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.

Check whether the statement is true or false

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.

Check whether the statement is true or false

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.

Check whether the statement is true or 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.

Check whether the statement is true or false

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.