Math 205B Test 1 (60 points)
Name:
•Check that you have 7 questions on two pages.
•Show all your work to receive full credit for a problem.
1. (10 points) (For this problem do all calculations by hand.) Determine all possible values of h
and ksuch that the solution set of the following system
(a) is empty (b) contains infinitely many solutions
2x1+hx2=5
4x1+3x2=k
2. (8 points) Let A=[~a 1~a 2~a 3~a 4]bea3×4 matrix. Suppose x1=3,x
2=4,x
3=1,x
4=2is
a solution of the equation A~x=~
0.
(a) Are the columns of Alinearly independent? Explain.
(b) Write the vector ~a 4as a linear combination of the vectors ~a 1,~a 2, and ~a 3.
