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Homework 5
- Section 4.1 Definition of the Determinant
- Read Section 4.1. Do the True/False questions (pp. 255-256). Do Problem 4.1(a)(c)(e)(g)(i). WeBWorK reading question due March 22 @ 9:50 am.
- In class (group work): 4.1(f), 4.7. Final due date: April 3
- Homework: 4.1(b)(d)(h)(j), 4.4, 4.5, 4.8, 4.9, 4.10 (Use the result of Problem 4.9)
- Hand In: 4.1(h)*, 4.10(b)*
- 4.1(h)*. Complete Problem 4.1(h) on Page 256, but with the matrix \[\left|\matrix{-2&2&3\cr 1&4&2\cr 6&5&-3\cr}\right|\]
- 4.10(b)*. Similar to Problem 4.10(b) on Page 258: Prove that \(Z^t(X\times Y) = X^t(Y\times Z)\). (Use the result of Problem 4.9)
- WeBWorK assignment for Sections 4.1-4.2 due March 31 @ 11 pm
- Read Section 4.2. Do the True/False questions (p. 265). Do Problems 4.12(a)(c)(e). WeBWorK reading question due March 24 @ 9:50 am.
- In class (group work): 4.12(b), 4.16. Final due date: April 5
- Homework: 4.12(d)(f), 4.14, 4.15, 4.17, 4.23, 4.24, 4.25, 4.26
- Hand In: 4.12(d)*, 4.24, 4.26
- 4.12(d)*. Complete Problem 4.12(d) on Page 266, but with the matrix \[\left|\matrix{2&1&2&0\cr 2&1&2&1\cr 0&0&3&1\cr 4&3&2&5\cr}\right|\]
- 4.24, as in the text
- 4.26, as in the text
- Read Section 4.3. Do the True/False questions (p. 275). Do Problem 4.28. WeBWorK reading question due March 29 @ 9:50 am.
- In class (group work): 4.29, 4.36. Final due date: April 12
- Homework: 4.30, 4.32, 4.35
- Hand In: 4.30*, 4.35*
- 4.30*. Complete Problem 4.30 on Page 275, but with the system \[\matrix{x+3y-2z&=&p_1\cr 2x+2y-z&=&p_2\cr 2x+3y+4z&=&p_3\cr}.\]
- 4.35*. Complete Problem 4.35 on Page 276, but with the matrix \[A=\left[\matrix{x^2&x^3&x^4\cr x&x&x^2\cr x^4&x^3&x\cr}\right].\] (Hint: det(\(A\))\(= x^4(1-x)(1-x^3)=x^4-x^5-x^7+x^8\).)
- Read Section 5.1. Do the True/False questions (pp. 287-288). Do Problems 5.1, 5.2(a). WeBWorK reading question due March 31 @ 9:50 am.
- A Geogebra activity on eigenvectors
- In class (group work): 5.3(a), 5.5(f), 5.12. Final due date: April 17
- Homework: 5.2, 5.3(b)(c), 5.5(d)(h), 5.10 (do not use decimal approximations while solving this problem - use exact values (e.g., a square root of a number, not its decimal approximation)), 5.11, 5.13, 5.14
- Hand In: 5.3(c)*, 5.5(h)*, 5.10* (do not use decimal approximations while solving this problem - use exact values (e.g., a square root of a number, not its decimal approximation))
- 5.3(c)*. Complete Problem 5.3(c) on Page 288, but with the matrix \[\left[\matrix{0&-1&-1\cr 0&1&0\cr -1&-1&0\cr}\right]\] and the vectors \[\left[\matrix{1\cr 0\cr 1\cr}\right], \left[\matrix{-1\cr 0\cr 1\cr}\right], \left[\matrix{0\cr -1\cr 1\cr}\right].\]
- 5.5(h)*. Complete Problem 5.5(h) on Page 289, but with the matrix \[\left[\matrix{2&1&0\cr 4&1&3\cr -1&-1&0\cr}\right].\] Note the problem asks for eigenvalues and eigenvectors over \(\mathbb{R}\).
- 5.10*. Similar to Problem 5.10 on Page 290. The Lucas sequence \(L_n\) is defined by \(L_1=2, L_2=1\), and \(L_{n+1}=L_n+L_{n-1}\) for \(n\geq 2\). Thus,
\[L_3=L_2+L_1=1+2=3, L_4=L_3+L_2=3+1=4\]
(a) Compute \(L_n\) for \(n=5, 6, 7\).
(b) Let \(A\) and \(X\) be as given below. Compute \(A^nX\) for \(n=1, 2, 3, 4, 5, 6\). What do you observe? Can you explain why this happens? (Give a proof) \[A=\left[\matrix{0&1\cr 1&1}\right],\quad X=\left[\matrix{2\cr 1}\right]\] (c) (Same as 5.10(c)) Find the eigenvalues and a basis for the corresponding eigenspaces for \(A\) and express \(X\) as a linear combination of the basis elements.
(d) Use the expression in Part (c) to compute \(A^8X\). Use this answer to find \(L_{10}\).
(e) Give a general formula for \(L_n\)
(Do not use decimal approximations while solving this problem - use exact values (e.g., a square root of a number, not its decimal approximation))
- Read Section 5.2. Do the True/False questions (p. 299). Do Problems 5.29(a)-(d). WeBWorK reading question due April 3 @ 9:50 am.
- In class (group work): 5.29(g) (use the characteristic polynomial provided), 5.33. Final due date: April 19
- Homework: 5.29(e)(h) (use the characteristic polynomials provided), 5.34, 5.35, 5.36, 5.37, 5.38
- Hand In: 5.29(h)* (use the characteristic polynomial provided), 5.38(a)*
- 5.29(h)*. Complete Problem 5.29(h) on Page 300, but with the matrix \[A=\left[\matrix{4&-2&2\cr 6&-3&4\cr 3&-2&3}\right].\] You may take as given that the characteristic polynomial of \(A\) is \(p_A(\lambda) = -(\lambda-2)(\lambda-1)^2\).
- 5.38(a)*. Complete Problem 5.38(a) on Page 301, but where \(A\) is the matrix \[\left[\matrix{1&a&1&b\cr 0&2&-1&c\cr 0&0&1&d\cr 0&0&0&2}\right].\]