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Chapter 8: Q8.1-13E (page 437)

Question: 13. Suppose \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a basis for \({\mathbb{R}^3}\). Show that Span \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\) is a plane in \({\mathbb{R}^3}\). (Hint: What can you say about \({\rm{u}}\) and \({\rm{v}}\)when Span \(\left\{ {{\rm{u,v}}} \right\}\) is a plane?)

Short Answer

Expert verified

It is shown that the Span of \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\)is in the plane \({\mathbb{R}^3}\).

Step by step solution

01

Describe the given statement

It is given that \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a basis for \({\mathbb{R}^3}\)and it is to be shown that Span of \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\)is in the plane \({\mathbb{R}^3}\). This is possible only if the set of vectors \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\)is linearly independent.

02

 Make a linear combination of the given set of vectors

A possible linear combination of a set of vectors \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\)is \({c_1}\left( {{{\bf{v}}_2} - {{\bf{v}}_1}} \right) + {c_2}\left( {{{\bf{v}}_3} - {{\bf{v}}_1}} \right)\), where \({c_2},{c_3}\) satisfies \({c_1}\left( {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}}} \right) + {c_2}\left( {{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right) = 0\). If \({c_1}\left( {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}}} \right) + {c_2}\left( {{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right) = 0\) is linearly independent, then \({c_1}\left( {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}}} \right) + {c_2}\left( {{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right) = 0\) must be equal to 0, which is possible only if \({c_1} = {c_2} = 0\).

03

 Use property of a linear independent set

According to the property of linear independent set, if \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a set of linearly independent vectors, then all the coefficients of its linear combination are 0. So, \({c_1} = {c_2} = 0\) is true for all \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) in \({\mathbb{R}^3}\).

04

 Draw a conclusion

As the set of vectors \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\)is linearly independent, so the Span of \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\)is in the plane \({\mathbb{R}^3}\).

Hence, it is proved that the Span of \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\)is in the plane \({\mathbb{R}^3}\).

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Most popular questions from this chapter

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left( {f:d} \right)\) the hyperplane H described in the exercise.

Let H be the plane in \({\mathbb{R}^{\bf{3}}}\) spanned by the rows of \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{\bf{5}}\\{\bf{0}}&{\bf{2}}&{\bf{4}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\). (Hint: How is H is related to Nul B?see section 6.1.)

Let\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)be an affinely dependent set of points in\({\mathbb{R}^{\bf{n}}}\)and let\(f:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\)be a linear transformation. Show that\(\left\{ {f\left( {{{\bf{p}}_1}} \right),f\left( {{{\bf{p}}_2}} \right),f\left( {{{\bf{p}}_3}} \right)} \right\}\)is affinely dependent in\({\mathbb{R}^{\bf{m}}}\).

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left( {f:d} \right)\) the hyperplane H described in the exercise.

Let H be the column space of the matrix \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}\\{\bf{5}}&{\bf{2}}\\{ - {\bf{4}}}&{ - {\bf{4}}}\end{array}} \right)\). That is, \(H = {\bf{Col}}\,B\).(Hint: How is \({\bf{Col}}\,B\)related to Nul \({B^T}\)? See section 6.1)

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left) {f:d} \right)\) the hyperplane H described in the exercise.

Let H be the plane in \({\mathbb{R}^{\bf{3}}}\) spanned by the rows of \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{4}}&{ - {\bf{5}}}\\{\bf{0}}&{ - {\bf{2}}}&{\bf{8}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\).

In Exercises 21–24, a, b, and c are noncollinear points in\({\mathbb{R}^{\bf{2}}}\)and p is any other point in\({\mathbb{R}^{\bf{2}}}\). Let\(\Delta {\bf{abc}}\)denote the closed triangular region determined by a, b, and c, and let\(\Delta {\bf{pbc}}\)be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\)is positive, where\(\overrightarrow {\bf{a}} \),\(\overrightarrow {\bf{b}} \)and\(\overrightarrow {\bf{c}} \)are the standard homogeneous forms for the points.

24. Take q on the line segment from b to c and consider the line through q and a, which may be written as\(p = \left( {1 - x} \right)q + xa\)for all real x. Show that, for each x,\(det\left[ {\begin{array}{*{20}{c}}{\widetilde p}&{\widetilde b}&{\widetilde c}\end{array}} \right] = x \cdot det\left[ {\begin{array}{*{20}{c}}{\widetilde a}&{\widetilde b}&{\widetilde c}\end{array}} \right]\). From this and earlier work, conclude that the parameter x is the first barycentric coordinate of p. However, by construction, the parameter x also determines the relative distance between p and q along the segment from q to a. (When x = 1, p = a.) When this fact is applied to Example 5, it shows that the colors at vertex a and the point q are smoothly interpolated as p moves along the line between a and q.
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