HOMEWORK
Assignment 1
Problem 1.1.1c, 1.1.2ab, 1.1.4b, 1.1.5, 1.1.6f, 1.1.7a, 1.1.8a, 1.1.9
Problem 1.2.1b, 1.2.3, 1.2.7, 1.2.8, 1.2.11, 1.2.13, 1.2.15, 1.2.16
Assignment 2
Problem 1.2.17, 1.2.20, 1.2.23
Problem 1.3.4, 1.3.9, 1.3.11, 1.3.19
Problem 1.4.2, 1.4.3, 1.4.5, 1.4.7
Assignment 3
Problem 1.3.14
Problem 1.4.8, 1.4.9b, 1.4.10a, 1.4.12
Problem 1.5.1, 1.5.3, 1.5.4, 1.5.6,1.5.10, 1.5.14
Assignment 4
Problem 1.5.9, 1.5.13, 1.5.16, 1.5.19, 1.5.24
Problem 1.6.2, 1.6.3, 1.6.5
Problem 1.7.4bd, 1.7.11bc, 1.7.14
Assignment 5
Problem 1.5.11, 1.5.14bc
Problem 1.7.6, 1.7.10, 1.7.15, 1.7.16, 1.7.18, 1.7.19
Problem 1.8.2, 1.8.7
Assignment 6
Problem 1.7.11ab, 1.7.14, 1.7.21, 1.7.22
Problem 1.8.8, 1.8.9, 1.8.11
Problem 1.9.1, 1.9.2ac
Assignment 7
Problem 1.8.6, 1.8.10, 1.9.3
Problem 2.1.2, 2.1.3cd, 2.1.8
Problem 2.2.1, 2.2.7, 2.2.9, 2.2.11
Problem 2.3.2efg, 2.3.4
Assignment 8
Problem 2.3.7, 2.3.8, 2.3.9, 2.3.11
Problem 2.4.2, 2.4.4, 2.4.5, 2.4.6
Problem 2.5.1a, 2.5.2, 2.5.3
Assignment 9
In preparationProblem 2.4.7, 2.4.8, 2.4.13
Problem 2.5.6, 2.5.7, 2.5.8
Problem 2.6.1, 2.6.3, 2.6.4
Problem 2.7.1, 2.7.2, 2.7.3, 2.7.4
Assignment 10
Problem 4.8.1(use any method), 4.8.4, 4.8.11, 4.8.15, 4.8.18b,
Let A be an n by n matrix with real entries. Let b be a real number. Show that det (bA) = (b^n)detA.
Let A be an n by n matrix such that A = -A' where A' is the transpose of A. If n is odd, prove A cannot be invertible. Find an example for A with n even satisfying the previous hypothesis and det A not equal to 0.
Let C be the n by n matrix with rows C^1 = e_n, C^2 = e_n + e_{n-1}, ...C^n = e_n + e_{n-1} + ... e_1. Here the e_i are the standard basis of R^n. Find det C.
Let P=(x_1, y_1) and Q=(x_2, y_2) be two points in R^2, which are not collinear with the origin. Show that the equation of the line in R^2 determined by PQ is given by det(S)=0 where the S is the 3 by 3 matrix with columns (x_1, y_1, 1), (x_2, y_2, 1), (x,y,1).
Let A be an n by n real matrix. Let B be an m by m real matrix with m greater than n. Let C be an m by n real matrix. Form D a real m+n by m+n matris whose first n rows are (A,0) where 0 denotes the n by m matrix with all entries 0. Let the bottom m rows of D consist of (C,B). Show det D = (detA)(detB).
Assignment 11
Problem 2.10.1bde, 2.10.2, 2.10.4, 2.10.5, 2.10.6, 2.10.9, 2.10.10
Problem 3.6.2a, 3.6.5a, 3.6.6, 3.6.7a
Assignment 12
Problem 3.1.5, 3.1.7, 3.1.10, 3.1.11, 3.1.21, 3.1.24, 3.1.25
Problem 3.2.3, 3.2.6, 3.2.7, 3.2.9, 3.2.12
Assignment 13
Problem 3.3.5, 3.3.12, 3.3.13
Problem 3.4,1, 3.4.2, 3.4.3, 3.4.4, 3.4.7, 3.4.8, 3.4.9, 3.4.11
Assignment 14
Problem 3.5.3, 3.5.4, 3.5.5, 3.5.8, 3.5.11, 3.5.12
Problem 3.6.1, 3.6.2, 3.6.5, 3.6.7, 3.6.8
Notes
Jacob raised in class today 12/11 the question of what happens if H is a real symmetric matrix with first row (a,b) and bottom row (b,d) with a and d having opposite signs and det H > 0. Note that the condition det H > 0 makes this impossible.
Problems: find critical points and classify in all cases:
1) f(y,z) = (z^3 -3z)/(1+y^2)
2) h(x,y,z) = (3x^4 - 6x^2 +1)/(1+ z^2) - y^2
3) g(x,y,z) = x^3 - 3x +y^2 + z^2
4) k(x,y,z) = (1 + 2x + 3y -z)^2
I might add a few more.