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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

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\({\mathbb{R}^{\bf{3}}}\).

Short Answer

Expert verified

The coordinates of this site relative to the standard basis of \({\mathbb{R}^{\bf{3}}}\) is \({\bf{x}} = \left( {\begin{array}{*{20}{c}}{1.3}\\0\\{0.8}\end{array}} \right)\).

Step by step solution

01

State the coordinates of x relative to the standard basis

It is given that for the unit cell, thebasis is\(B = \left\{ {\left( {\begin{array}{*{20}{c}}{2.6}\\{ - 1.5}\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\3\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\0\\{4.8}\end{array}} \right)} \right\}\).

Obtain thecoordinatesof the octahedral site \({\left( {\bf{x}} \right)_B} = \left( {\begin{array}{*{20}{c}}{1/2}\\{1/4}\\{1/6}\end{array}} \right)\) relative to the standard basis of\({\mathbb{R}^{\bf{3}}}\), as shown below:

Write the basis B in the matrix form as shown below:

\({P_B} = \left( {\begin{array}{*{20}{c}}{2.6}&0&0\\{ - 1.5}&3&0\\0&0&{4.8}\end{array}} \right)\)

02

State the coordinates of x relative to the standard basis

Compute the product of matrices\({\left( {\bf{x}} \right)_B} = \left( {\begin{array}{*{20}{c}}{1/2}\\{1/4}\\{1/6}\end{array}} \right)\)and\({P_B} = \left( {\begin{array}{*{20}{c}}{2.6}&0&0\\{ - 1.5}&3&0\\0&0&{4.8}\end{array}} \right)\)using the MATLAB command shown below:

\(\begin{array}{l} > > {\rm{A}} = \left( {{\rm{2}}{\rm{.6 0 0; }} - {\rm{1}}{\rm{.5 3 0; 0 0 4}}{\rm{.8}}} \right);\\ > > {\rm{B}} = \left( {{\rm{1/2 1/4 1/6}}} \right);\\ > > {\rm{M}} = {\rm{A}}*{\rm{B}}\end{array}\)

So, the output is

\(\begin{array}{c}{\bf{x}} = {P_B}{\left( {\bf{x}} \right)_B}\\ = \left( {\begin{array}{*{20}{c}}{2.6}&0&0\\{ - 1.5}&3&0\\0&0&{4.8}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{1/2}\\{1/4}\\{1/6}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{1.3}\\0\\{0.8}\end{array}} \right).\end{array}\)

Thus, the coordinates of this site relative to the standard basis of \({\mathbb{R}^{\bf{3}}}\) is \({\bf{x}} = \left( {\begin{array}{*{20}{c}}{1.3}\\0\\{0.8}\end{array}} \right)\).

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Most popular questions from this chapter

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

20. \(A = \left( {\begin{array}{*{20}{c}}{.8}&{ - .3}&0\\{.2}&{.5}&1\\0&0&{ - .5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\1\\0\end{array}} \right)\).

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

17. A submatrix of a matrix A is any matrix that results from deleting some (or no) rows and/or columns of A. It can be shown that A has rank \(r\) if and only if A contains an invertible \(r \times r\) submatrix and no longer square submatrix is invertible. Demonstrate part of this statement by explaining (a) why an \(m \times n\) matrix A of rank \(r\) has an \(m \times r\) submatrix \({A_1}\) of rank \(r\), and (b) why \({A_1}\) has an invertible \(r \times r\) submatrix \({A_2}\).

The concept of rank plays an important role in the design of engineering control systems, such as the space shuttle system mentioned in this chapterโ€™s introductory example. A state-space model of a control system includes a difference equation of the form

\({{\mathop{\rm x}\nolimits} _{k + 1}} = A{{\mathop{\rm x}\nolimits} _k} + B{{\mathop{\rm u}\nolimits} _k}\)for \(k = 0,1,....\) (1)

Where \(A\) is \(n \times n\), \(B\) is \(n \times m\), \(\left\{ {{{\mathop{\rm x}\nolimits} _k}} \right\}\) is a sequence of โ€œstate vectorsโ€ in \({\mathbb{R}^n}\) that describe the state of the system at discrete times, and \(\left\{ {{{\mathop{\rm u}\nolimits} _k}} \right\}\) is a control, or input, sequence. The pair \(\left( {A,B} \right)\) is said to be controllable if

\({\mathop{\rm rank}\nolimits} \left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}& \cdots &{{A^{n - 1}}B}\end{array}} \right) = n\) (2)

The matrix that appears in (2) is called the controllability matrix for the system. If \(\left( {A,B} \right)\) is controllable, then the system can be controlled, or driven from the state 0 to any specified state \({\mathop{\rm v}\nolimits} \) (in \({\mathbb{R}^n}\)) in at most \(n\) steps, simply by choosing an appropriate control sequence in \({\mathbb{R}^m}\). This fact is illustrated in Exercise 18 for \(n = 4\) and \(m = 2\). For a further discussion of controllability, see this textโ€™s website (Case study for Chapter 4).

Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?

In Exercise 18, Ais an \(m \times n\) matrix. Mark each statement True or False. Justify each answer.

18. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.

b. Row operations preserve the linear dependence relations among the rows of A.

c. The dimension of the null space of A is the number of columns of A that are not pivot columns.

d. The row space of \({A^T}\) is the same as the column space of A.

e. If A and B are row equivalent, then their row spaces are the same.

[M] Let \(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

  1. Construct matrices \(C\) and \(N\) whose columns are bases for \({\mathop{\rm Col}\nolimits} A\) and \({\mathop{\rm Nul}\nolimits} A\), respectively, and construct a matrix \(R\) whose rows form a basis for Row\(A\).
  2. Construct a matrix \(M\) whose columns form a basis for \({\mathop{\rm Nul}\nolimits} {A^T}\), form the matrices \(S = \left[ {\begin{array}{*{20}{c}}{{R^T}}&N\end{array}} \right]\) and \(T = \left[ {\begin{array}{*{20}{c}}C&M\end{array}} \right]\), and explain why \(S\) and \(T\) should be square. Verify that both \(S\) and \(T\) are invertible.
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