Math 54 Fall 2005 Quizzes

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Scott's sections

Week 2

Solve the system of equations q x - y = -q^{-2}, (q+q^{-1})x - y = 1, by writing down the matrix of coefficients, and inverting it. (Hint; you can check you've got a sensible solution by seeing if it stills works when you plug in q=1.)

An answer: x = q^{-1} + q, y = q^{-2} + 1 + q^2

Week 3

Can the matrix {{1,4,3},{2,4,2},{4,4,0}} be written as a product of elementary matrices? Why?

An answer: No; it's easy to show this square matrix is row equivalent to a matrix with a row of zeroes, and so can not be written as a product of elementary matrices. (Equivalently, it's singular, it's not invertible, etc.)

Week 4

I'm trying to make the set {A,B} into a vector space. I do this by defining A+A=A, A+B=B+A=B, B+B=A, and when x is a real number, and Z is either A or B, defining scalar multiplication by xZ = A, if x is zero, and xZ=Z if x is any other real number. What goes wrong?

An answer: Observe (5+7)B=12B=B, but 5B+7B=B+B=A. That is, one of the distributivity axioms fails.

Tom's sections

Quiz 2:

1) Let A=[0 -1 \\ 1 0]. Find all 2x2 matrices B such that AB=BA.

2) Does there exist a 2x2 A such that A^2=I, but A \neq [a 0 \\ 0 a]?

Quiz 4:

1) what is the span of (1,0,1), (2,1,1), (1,1,0). Is (2,1,-2) perpendicular to any of those three?

2) What is the span of two vectors u1 and u2 in R^n? If a nonzero vector w is perpendicular to both u1 and u2, can it be in span(u1,u2)?

(Most students gave a heuristic geometric argument. A few students looked at the projection of w onto ui, or wrote w as a linear combination and then looked at w.w).