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Homework 9
- Section 5.1 Eigenvectors
- Read Section 5.1. Do the True/False questions (pp. 287-288). Do Problems 5.1, 5.2(a).
- A Geogebra activity on eigenvectors
- 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: 5.5(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))
- 5.5(h)*. Complete Problem 5.5(h) on Page 289, but with the matrix \[\left[\matrix{3&0&0\cr -2&-1&2\cr -5&-10&8\cr}\right].\] 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).
- 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 \[A=\left[\matrix{6&-2&2\cr 6&-1&4\cr 3&-2&5}\right].\] You may take as given that the characteristic polynomial of \(A\) is \(p_A(\lambda) = -(\lambda-4)(\lambda-3)^2\).
- 5.38(a)*. Complete Problem 5.38(a) on Page 301, but where \(A\) is the matrix \[\left[\matrix{1&-1&1&b\cr 0&2&a&c\cr 0&0&1&d\cr 0&0&0&2}\right].\]
- Read Section 5.3. Do Problem 5.44.
- Homework: 5.46, 5.50, 5.54, 5.56, 5.57
- Hand In: 5.46*
- 5.46*. Complete Problem 5.46 on Page 313, but with the matrix \[A=\left[\matrix{1&0&-1\cr 0&1&0\cr 1&1&1\cr}\right]..\]