Question

Let

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{ {20}{c}}{\bf{2}}\\{\bf{0}}\\{ - {\bf{1}}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}{\bf{0}}\\{ - {\bf{2}}}\\{\bf{2}}\\{\bf{1}}\end{aligned}} \right)\), and let \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}{ - {\bf{2}}}\\{\bf{1}}\\{\bf{0}}\\{\bf{2}}\end{aligned}} \right)\)be the orthogonal set \(\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}},\,{{\bf{v}}_{\bf{3}}},\,{{\bf{v}}_{\bf{4}}}} \right\}\). Determine whether \({{\bf{p}}_i}\) is in span S, \({\bf{aff}}\,S\), or \({\bf{conv}}\,\,S\).

a. \({{\bf{p}}_{\bf{1}}}\) b. \({{\bf{p}}_{\bf{2}}}\) c. \({{\bf{p}}_{\bf{3}}}\) d. \({{\bf{p}}_{\bf{4}}}\)

Short answer

\({{\bf{p}}_1} \notin {\rm{conv}}\,\,S\) and \({{\bf{p}}_2} \in {\rm{conv}}\,\,S\).

Step-by-step solution

Find the projection of \({{\bf{p}}_{\bf{1}}}\), \({{\bf{p}}_{\bf{2}}}\), and \({{\bf{p}}_{\bf{3}}}\)

As S is an orthogonal set, the projection of \({{\bf{p}}_1}\) is:

\({\rm{pro}}{{\rm{j}}_w}{{\bf{p}}_1} = \frac{{{{\bf{p}}_1} \cdot {{\bf{v}}_1}}}{{{{\bf{v}}_1} \cdot {{\bf{v}}_1}}}{{\bf{v}}_1} + \frac{{{{\bf{p}}_1} \cdot {{\bf{v}}_2}}}{{{{\bf{v}}_2} \cdot {{\bf{v}}_2}}}{{\bf{v}}_2} + \frac{{{{\bf{p}}_1} \cdot {{\bf{v}}_3}}}{{{{\bf{v}}_3} \cdot {{\bf{v}}_3}}}{{\bf{v}}_3}\)

The projection of \({{\bf{p}}_{\bf{2}}}\) is:

\({\rm{pro}}{{\rm{j}}_w}{{\bf{p}}_2} = \frac{{{{\bf{p}}_2} \cdot {{\bf{v}}_1}}}{{{{\bf{v}}_1} \cdot {{\bf{v}}_1}}}{{\bf{v}}_1} + \frac{{{{\bf{p}}_2} \cdot {{\bf{v}}_2}}}{{{{\bf{v}}_2} \cdot {{\bf{v}}_2}}}{{\bf{v}}_2} + \frac{{{{\bf{p}}_2} \cdot {{\bf{v}}_3}}}{{{{\bf{v}}_3} \cdot {{\bf{v}}_3}}}{{\bf{v}}_3}\)

The projection of \({{\bf{p}}_3}\) is:

\({\rm{pro}}{{\rm{j}}_w}{{\bf{p}}_3} = \frac{{{{\bf{p}}_3} \cdot {{\bf{v}}_1}}}{{{{\bf{v}}_1} \cdot {{\bf{v}}_1}}}{{\bf{v}}_1} + \frac{{{{\bf{p}}_3} \cdot {{\bf{v}}_2}}}{{{{\bf{v}}_2} \cdot {{\bf{v}}_2}}}{{\bf{v}}_2} + \frac{{{{\bf{p}}_3} \cdot {{\bf{v}}_3}}}{{{{\bf{v}}_3} \cdot {{\bf{v}}_3}}}{{\bf{v}}_3}\)

The projection of \({{\bf{p}}_4}\) is:

\({\rm{pro}}{{\rm{j}}_w}{{\bf{p}}_4} = \frac{{{{\bf{p}}_4} \cdot {{\bf{v}}_1}}}{{{{\bf{v}}_1} \cdot {{\bf{v}}_1}}}{{\bf{v}}_1} + \frac{{{{\bf{p}}_4} \cdot {{\bf{v}}_2}}}{{{{\bf{v}}_2} \cdot {{\bf{v}}_2}}}{{\bf{v}}_2} + \frac{{{{\bf{p}}_4} \cdot {{\bf{v}}_3}}}{{{{\bf{v}}_3} \cdot {{\bf{v}}_3}}}{{\bf{v}}_3}\)

Find the product in the projection of \({{\bf{p}}_{\bf{1}}}\)

The product for projection \({{\bf{p}}_{\bf{1}}}\):

\(\begin{aligned}{c}{{\bf{p}}_{\bf{1}}} \cdot {{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{ {20}{c}}{ - 1}\\2\\{ - \frac{3}{2}}\\{\frac{5}{2}}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right)\\ = \frac{9}{2}\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{1}}} \cdot {{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right)\\ = 9\end{aligned}\)

And

\(\begin{aligned}{c}{{\bf{p}}_{\bf{1}}} \cdot {{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}{ - 1}\\2\\{ - \frac{3}{2}}\\{\frac{5}{2}}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right)\\ = - \frac{9}{2}\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{2}}} \cdot {{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right)\\ = 9\end{aligned}\)

And

\(\begin{aligned}{c}{{\bf{p}}_{\bf{1}}} \cdot {{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}{ - 1}\\2\\{ - \frac{3}{2}}\\{\frac{5}{2}}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right)\\ = 9\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{3}}} \cdot {{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right)\\ = 9\end{aligned}\)

Find the product in the projection of \({{\bf{p}}_{\bf{2}}}\)

The product for projection of \({{\bf{p}}_2}\) is:

\(\begin{aligned}{c}{{\bf{p}}_2} \cdot {{\bf{v}}_2} = \left( {\begin{aligned}{ {20}{c}}{ - \frac{1}{2}}\\0\\{\frac{1}{4}}\\{\frac{7}{4}}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right)\\ = \frac{9}{4}\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{1}}} \cdot {{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right)\\ = 9\end{aligned}\)

And,

\(\begin{aligned}{c}{{\bf{p}}_2} \cdot {{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}{ - \frac{1}{2}}\\0\\{\frac{1}{4}}\\{\frac{7}{4}}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right)\\ = \frac{9}{4}\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{2}}} \cdot {{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right)\\ = 9\end{aligned}\)

And,

\(\begin{aligned}{c}{{\bf{p}}_2} \cdot {{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}{ - \frac{1}{2}}\\0\\{\frac{1}{4}}\\{\frac{7}{4}}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right)\\ = \frac{9}{2}\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{3}}} \cdot {{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right)\\ = 9\end{aligned}\)

Find the product in the projection of \({{\bf{p}}_{\bf{3}}}\)

The product for projection of \({{\bf{p}}_3}\) is:

\(\begin{aligned}{c}{{\bf{p}}_3} \cdot {{\bf{v}}_2} = \left( {\begin{aligned}{ {20}{c}}6\\{ - 4}\\1\\{ - 1}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right)\\ = 9\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{1}}} \cdot {{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right)\\ = 9\end{aligned}\)

And,

\(\begin{aligned}{c}{{\bf{p}}_3} \cdot {{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}6\\{ - 4}\\1\\{ - 1}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right)\\ = 9\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{2}}} \cdot {{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right)\\ = 9\end{aligned}\)

And,

\(\begin{aligned}{c}{{\bf{p}}_3} \cdot {{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}6\\{ - 4}\\1\\{ - 1}\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right)\\ = - 18\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{3}}} \cdot {{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right)\\ = 9\end{aligned}\)

Find the product in the projection of \({{\bf{p}}_{\bf{4}}}\)

The product for projection of \({{\bf{p}}_3}\) is:

\(\begin{aligned}{c}{{\bf{p}}_4} \cdot {{\bf{v}}_2} = \left( {\begin{aligned}{ {20}{c}}{ - 1}\\{ - 2}\\0\\4\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right)\\ = 6\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{1}}} \cdot {{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}2\\0\\{ - 1}\\2\end{aligned}} \right)\\ = 9\end{aligned}\)

And,

\(\begin{aligned}{c}{{\bf{p}}_4} \cdot {{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}{ - 1}\\{ - 2}\\0\\4\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right)\\ = 8\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{2}}} \cdot {{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}0\\{ - 2}\\2\\1\end{aligned}} \right)\\ = 9\end{aligned}\)

And,

\(\begin{aligned}{c}{{\bf{p}}_4} \cdot {{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}{ - 1}\\{ - 2}\\0\\4\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right)\\ = 8\end{aligned}\), \(\begin{aligned}{c}{{\bf{v}}_{\bf{3}}} \cdot {{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right) \cdot \left( {\begin{aligned}{ {20}{c}}{ - 2}\\1\\0\\2\end{aligned}} \right)\\ = 9\end{aligned}\)

Substitute values in the equation of projections

The projection \({{\bf{p}}_1}\) is:

\(\begin{aligned}{c}{\rm{pro}}{{\rm{j}}_w}{{\bf{p}}_1} = \frac{{\frac{9}{2}}}{9}{{\bf{v}}_1} + \frac{{\left( { - \frac{9}{2}} \right)}}{9}{{\bf{v}}_2} + \frac{9}{9}{{\bf{v}}_3}\\ = \frac{1}{2}{{\bf{v}}_1} - \frac{1}{2}{{\bf{v}}_2} + {{\bf{v}}_3}\end{aligned}\)

The projection \({{\bf{p}}_2}\) is:

\(\begin{aligned}{c}{\rm{pro}}{{\rm{j}}_w}{{\bf{p}}_2} = \left( {\frac{{\frac{9}{4}}}{9}} \right){{\bf{v}}_1} + \left( {\frac{{\frac{9}{4}}}{9}} \right){{\bf{v}}_2} + \left( {\frac{{\frac{9}{2}}}{9}} \right){{\bf{v}}_3}\\ = \frac{1}{4}{{\bf{v}}_1} + \frac{1}{4}{{\bf{v}}_2} + \frac{1}{2}{{\bf{v}}_3}\end{aligned}\)

The projection \({{\bf{p}}_3}\) is:

\(\begin{aligned}{c}{\rm{pro}}{{\rm{j}}_w}{{\bf{p}}_3} = \left( {\frac{9}{9}} \right){{\bf{v}}_1} + \left( {\frac{9}{9}} \right){{\bf{v}}_2} + \left( {\frac{{ - 18}}{9}} \right){{\bf{b}}_3}\\ = {{\bf{v}}_1} + {{\bf{v}}_2} - 2{{\bf{v}}_3}\end{aligned}\)

The projection \({{\bf{p}}_4}\) is:

\(\begin{aligned}{c}{\rm{pro}}{{\rm{j}}_w}{{\bf{p}}_4} = \frac{6}{9}{{\bf{v}}_1} + \frac{8}{9}{{\bf{v}}_2} + \frac{8}{9}{{\bf{v}}_3}\\ = \frac{2}{3}{{\bf{v}}_1} + \frac{8}{9}{{\bf{v}}_2} + \frac{8}{9}{{\bf{v}}_3}\end{aligned}\)

Following projections can be made from the projections:

1. For \({{\bf{p}}_1}\), sm of coefficients is 1, so \({{\bf{p}}_1}\) is affine of S, but all coefficients are not positive therefore \({{\bf{p}}_1}\) is not conv of S. \({{\bf{p}}_1}\) is a span of S.
2. For \({{\bf{p}}_2}\), the sum of coefficients is 1, so \({{\bf{p}}_1}\) is affine of S and all coefficients are positive, so \({{\bf{p}}_2}\) is conv S. \({{\bf{p}}_2}\) is a span of S.
3. For \({{\bf{p}}_3}\), the sum of coefficients is not 1, so \({{\bf{p}}_3}\) is not affine of S. As all coefficients are not positive, \({{\bf{p}}_3}\) is not in conv S. \({{\bf{p}}_2}\) is a span of S.
4. For \({{\bf{p}}_4}\), the sum of coefficients is not 1, so \({{\bf{p}}_3}\) is not affine of S. As all coefficients are not positive, \({{\bf{p}}_3}\) is not in conv S. \({{\bf{p}}_2}\) is a span of S.