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Chapter 4: Q24E (page 191)

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\) and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Show that the coordinate mapping is onto \({\mathbb{R}^n}\). That is, given any y in \({\mathbb{R}^n}\), with entries \({y_{\bf{1}}}\),鈥�.,\({y_n}\), produce u in V such that \({\left( {\bf{u}} \right)_B} = y\).

Short Answer

Expert verified

The given mapping is onto.

Step by step solution

01

Find the coordinate u

\(y\)in \({\mathbb{R}^n}\) is expressed as:

\(y = \left( {\begin{array}{*{20}{c}}{{y_1}}\\{{y_2}}\\{{y_3}}\end{array}} \right)\)

The coordinates of \({\bf{u}}\) relative to basis B are the weights \({y_1}\), \({y_2}\),鈥� \({y_n}\) such that

\({\bf{u}} = {y_1}{{\bf{b}}_1} + {y_2}{{\bf{b}}_2} + .... + {y_n}{{\bf{b}}_n}\)

02

Find the vector \({\left( {\bf{u}} \right)_B}\)

The coordinate of vector u is:

\(\begin{array}{c}{\left( {\bf{u}} \right)_B} = \left( {\begin{array}{*{20}{c}}{{y_1}}\\{{y_2}}\\{{y_3}}\end{array}} \right)\\ = y\end{array}\)

For any arbitrary element \(y \in {\mathbb{R}^n}\), there is an element u in V. So, every vector in \({\mathbb{R}^n}\) has a pre-image corresponding to the mapping.

Thus, the coordinate mapping is onto.

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Most popular questions from this chapter

A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Discuss.

If A is a \({\bf{7}} \times {\bf{5}}\) matrix, what is the largest possible rank of A? If Ais a \({\bf{5}} \times {\bf{7}}\) matrix, what is the largest possible rank of A? Explain your answer.

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

16.

  1. If\(B\)is the standard basis for\({\mathbb{R}^n}\)then the\(B\)-coordinate vector of an\({\mathop{\rm x}\nolimits} \)in\({\mathbb{R}^n}\)is x itself.
  2. The correspondence\({\left( {\mathop{\rm x}\nolimits} \right)_B} \mapsto {\mathop{\rm x}\nolimits} \)is called coordinate mapping.
  3. In some cases, a plane in\({\mathbb{R}^3}\)can be isomorphic to\({\mathbb{R}^2}\).

Exercises 37 and 38 concern the crystal lattice for titanium, which has the hexagonal structure shown on the left in the accompanying

figure. The vectors\(\left( {\begin{array}{*{20}{c}}{2.6}\\{ - 1.5}\\0\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}0\\3\\0\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}0\\0\\{4.8}\end{array}} \right)\)in\({\mathbb{R}^{\bf{3}}}\)form a basis for the unit cell shown on the right. The numbers here are Angstrom units\(\left( {1\mathop { A}\limits^{{\rm{ o}}} = 1{0^{ - 8}}cm} \right)\). In alloys of titanium, some additional atoms may be in the unit cell at the octahedral and tetrahedralsites (so named because of the geometric objects

formed by atoms at these locations).


The hexagonal close-packed lattice and its unit cell.

37. One of the octahedral sites is\(\left( {\begin{array}{*{20}{c}}{1/2}\\{1/4}\\{1/6}\end{array}} \right)\), relative to the lattice basis. Determine the coordinates of this site relative to the standard basis of\({\mathbb{R}^{\bf{3}}}\).

In Exercise 4, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

4. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{0}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\\{\bf{2}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{3}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{8}}\\{ - {\bf{7}}}\end{array}} \right)\)
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