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Chapter 8: Q13E (page 437)

Question: a. Determine the number of k-faces of the 5-dimensional hypercube \({C^{\bf{5}}}\) for \(k = {\bf{0}},{\bf{1}},.....,{\bf{4}}\). Verfy that your answer satisfies Euler鈥檚 formula.

b. Make a chart of the values of \({f_k}\left( {{C^n}} \right)\) for \(n = {\bf{1}},.....,{\bf{5}}\) and \(k = {\bf{0}},{\bf{1}},.....,{\bf{4}}\). Can you see a pattern? Guess a general formula for \({f_k}\left( {{C^n}} \right)\).

Short Answer

Expert verified

a. 32, 80, 80, 40, 10

b.

\({f_0}\)

\({f_1}\)

\({f_3}\)

\({f_4}\)

\({f_5}\)

\({S^1}\)

2

\({S^2}\)

4

4

\({S^3}\)

8

12

6

\({S^4}\)

16

32

24

8

\({S^5}\)

32

80

80

40

10

There exist a pattern for \({f_k}\left( {{S^n}} \right)\) and it is given by the formula, \({f_k}\left( {{C^n}} \right) = {2^{k + 1}}\left( {\begin{array}{*{20}{c}}n\\k\end{array}} \right)\). Here, \(\left( {\begin{array}{*{20}{c}}a\\b\end{array}} \right) = \frac{{a!}}{{b!\left( {a - b} \right)!}}\) is the binomial coefficient.

Step by step solution

01

Find solution for part (a)

The number offaces for \(k = 0\):

\({f_0}\left( {{C^5}} \right) = 32\)

The number of faces for \(k = 1\):

\({f_1}\left( {{C^5}} \right) = 80\)

The number of faces for \(k = 2\):

\({f_2}\left( {{C^5}} \right) = 80\)

The number of faces for \(k = 3\):

\({f_3}\left( {{C^5}} \right) = 40\)

The number of faces for \(k = 4\):

\({f_4}\left( {{C^5}} \right) = 10\)

02

Verify the Euler’s formula

Euler鈥檚 formulacan be verified as follows:

\(\begin{array}{c}{f_0}\left( {{C^5}} \right) - {f_1}\left( {{C^5}} \right) + {f_2}\left( {{C^5}} \right) - {f_3}\left( {{C^5}} \right) + {f_4}\left( {{C^5}} \right) = 32 - 80 + 80 - 40 + 10\\ = 2\end{array}\)

So, Euler鈥檚 formula is verified.

03

Find the solution for part (b)

The chart below represents the values of \({f_k}\left( {{C^n}} \right)\).

\({f_0}\)

\({f_1}\)

\({f_3}\)

\({f_4}\)

\({f_5}\)

\({S^1}\)

2

\({S^2}\)

4

4

\({S^3}\)

8

12

6

\({S^4}\)

16

32

24

8

\({S^5}\)

32

80

80

40

10

There exist a pattern for the values in the chart. The pattern for the above chart is given by the formula \({f_k}\left( {{C^n}} \right) = {2^{k + 1}}\left( {\begin{array}{*{20}{c}}n\\k\end{array}} \right)\). Here \(\left( {\begin{array}{*{20}{c}}a\\b\end{array}} \right) = \frac{{a!}}{{b!\left( {a - b} \right)!}}\) is the binomial coefficient.

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Most popular questions from this chapter

Repeat Exercise 25 with\({v_1} = \left[ {\begin{array}{*{20}{c}}1\\{\bf{2}}\\{ - {\bf{4}}}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{8}}\\{\bf{2}}\\{ - {\bf{5}}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{{\bf{10}}}\\{ - {\bf{2}}}\end{array}} \right]\), \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{0}}\\{\bf{8}}\end{array}} \right]\), and \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{.{\bf{9}}}\\{{\bf{2}}.{\bf{0}}}\\{ - {\bf{3}}.{\bf{7}}}\end{array}} \right]\).

Question: In Exercise 3, determine whether each set is open or closed or neither open nor closed.

3. a. \(\left\{ {\left( {x,y} \right):y > {\bf{0}}} \right\}\)

b. \(\left\{ {\left( {x,y} \right):x = {\bf{2}}\,\,\,and\,\,{\bf{1}} \le y \le {\bf{3}}} \right\}\)

c. \(\left\{ {\left( {x,y} \right):x = {\bf{2}}\,\,\,and\,\,{\bf{1}} < y < {\bf{3}}} \right\}\)

d. \(\left\{ {\left( {x,y} \right):xy = {\bf{1}}\,\,\,and\,\,x > {\bf{0}}} \right\}\)

e. \(\left\{ {\left( {x,y} \right):xy \ge {\bf{1}}\,\,\,and\,\,x > {\bf{0}}} \right\}\)


Prove Theorem 6 for an affinely independent set\(S = \left\{ {{v_1},...,{v_k}} \right\}\)in\({\mathbb{R}^{\bf{n}}}\). [Hint:One method is to mimic the proof of Theorem 7 in Section 4.4.]

Question: Let \({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{ - {\bf{1}}}\\{\bf{3}}\end{array}} \right)\), \({{\bf{n}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{\bf{4}}\\{\bf{2}}\end{array}} \right)\), and \({{\bf{n}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{3}}\\{\bf{1}}\\{\bf{5}}\end{array}} \right)\), let \({H_{\bf{1}}}\) be the hyperplane in \({\mathbb{R}^{\bf{4}}}\) through \({{\bf{p}}_{\bf{1}}}\) with normal \({{\bf{n}}_{\bf{1}}}\), and let \({H_{\bf{2}}}\) be the hyperplane through \({{\bf{p}}_{\bf{2}}}\) with normal \({{\bf{n}}_{\bf{2}}}\). Give an explicit description of \({H_{\bf{1}}} \cap {H_{\bf{2}}}\). (Hint: Find a point p in \({H_{\bf{1}}} \cap {H_{\bf{2}}}\) and two linearly independent vectors \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) that span a subspace parallel to the 2-dimensional flat \({H_{\bf{1}}} \cap {H_{\bf{2}}}\).)

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 A be the \({\bf{1}} \times {\bf{4}}\) matrix \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{2}}}\end{array}} \right)\) and let \(b = {\bf{5}}\). Let \(H = \left\{ {{\bf{x}}\,\,{\rm{in}}\,{\mathbb{R}^{\bf{4}}}:A{\bf{x}} = {\bf{b}}} \right\}\).
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