Math 260 Exam 1 Jerry L. Kazdan
Feb. 9, 2012 12:00 – 1:20
Directions This exam has 10 questions (10 points each). Closed book, no calculators or
computers– but you may use one 300 ×500 card with notes on both sides. Neatness counts.
1. Which of the following sets are linear spaces?
a) The points X= (x1, x2, x3) in R3with the property x1−2x3= 0 .
Solution: Yes – obviously.
b) The set of solutions xof Ax = 0, where Ais an m×nmatrix.
Solution: Yes – obviously.
c) The set of polynomials p(x) with R1
−1p(x) cos 2x dx = 0.
Solution: Yes – obviously.
d) The set of solutions y=y(t) of y00 + 4y0+y=x2−3. [Note: You are not being asked
to solve this differential equation. You are only being asked a more primitive question.]
Solution: No. The function y(x)≡0 does not satisfy this equation.
2. Let Sand Tbe linear spaces and L:S→Tbe a linear map. Say V1and V2are (distinct!)
solutions of the equations LX =Y1while Wis a solution of LX =Y2. Answer the following
in terms of V1,V2, and W.
a) Find some solution of LX = 2Y1−3Y2.
Solution: For instance X:= 2V1−3W
b) Find another solution (other than W) of LX =Y2.
Solution: For instance X:= V2−V1+W.
3. Say you have klinear algebraic equations in nvariables; in matrix form we write AX =Y.
Give a proof or counterexample for each of the following.
a) If n=kthere is always at most one solution.
Solution: Counterexample: A= 0 matrix.
b) If n>k, given any Yyou can always solve AX =Y.
Solution: Counterexample: Ais the zero matrix.
c) If n>k the nullspace of Ahas dimension greater than zero.
Solution: Yes, by the Rank Theorem: dim N(A) = n−dim I(A)≥n−k≥1
d) If n<k then for some Ythere is no solution of AX =Y.
Solution: Yes, by the Rank Theorem: k > n = dim I(A) + dim N(A)≥dim I(A) . Thus
the dimension of the image of Ais less than k.
e) If n<k the only solution of AX = 0 is X= 0.
Solution: Counterexample: let Abe the zero matrix.
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