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Chapter 4: Q16Q (page 191)

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\({\mathbb{R}^2}\).

Short Answer

Expert verified

a. The given statement is true.

b. The given statement is false.

c. The given statement is true.

Step by step solution

01

Determine whether the given statement is true or false

a)

The entries in vector\({\mathop{\rm x}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\6\end{array}} \right)\)are thecoordinates of\({\mathop{\rm x}\nolimits} \)relative to the standard basis \(\varepsilon = \left\{ {{{\mathop{\rm e}\nolimits} _1},{{\mathop{\rm e}\nolimits} _2}} \right\}\). If\(\varepsilon = \left\{ {{{\mathop{\rm e}\nolimits} _1},{{\mathop{\rm e}\nolimits} _2}} \right\}\), then\({\left( {\mathop{\rm x}\nolimits} \right)_\varepsilon } = {\mathop{\rm x}\nolimits} \).

Thus, statement (a) is true.

02

Determine whether the given statement is true or false

b)

Let\(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},{{\mathop{\rm b}\nolimits} _2},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\)be a basis forvector space\(V\). Then the coordinate mapping \({\mathop{\rm x}\nolimits} \mapsto {\left( {\mathop{\rm x}\nolimits} \right)_B}\)is one-to-one linear transformation from\(V\)onto\({\mathbb{R}^n}\).

Thus, statement (b) is false.

03

Determine whether the given statement is true or false

c)

The plane isisomorphic to\({\mathbb{R}^2}\)if it passes through theorigin, as shown in Example 7.

Thus, statement (c) is true.

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

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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

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