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 .
b) The set of solutions xof Ax = 0, where Ais an m×nmatrix.
c) The set of polynomials p(x) with R1
−1p(x) cos 2x dx = 0.
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.]
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.
b) Find another solution (other than W) of LX =Y2.
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.
b) If n>k, given any Yyou can always solve AX =Y.
c) If n>k the nullspace of Ahas dimension greater than zero.
d) If n<k then for some Ythere is no solution of AX =Y.
e) If n<k the only solution of AX = 0 is X= 0.
4. Find a real 2 ×2 matrix Asuch that A4=Ibut A26=I.
5. Find a quadratic polynomial p(x) that passes through the three points (−1,0) , (0,−1) , and
(2,3). [Don’t bother to “simplify” your answer.]
6. Let A:R3→R2and B:R2→R3be given matrices, and let C:= BA :R3→R3. Show that
Ccannot be invertible.
7. In R3, find the distance from the point P:= (1,1,0) to the plane x+ 2y−z= 0 .
1
