Math 205A Exam 2, page 1 March 5, 2004 INITIALS
1. Let Abe the matrix below. Recall that the null space of Ais the set of solutions of the
equation Ax=0. Recall also that the column space of Aconsists of the vectors bfor which
the equation Ax=bhas at least one solution.
A=
1 0 6 4 0
0 1 −320
0 0 0 0 1
4−3 33 10 0
1A. Which of the following column vectors b1and b2is in the column space of A? Hint:
I’d row reduce the matrix [A|b1|b2] all at once, rather than use two separate augmented
matrices. Explain your answers!
b1=
4
−2
5
22
b2=
4
−2
5
18
1B. Find all the vectors in the null space of A, writing them in terms of a linear combi-
nation of appropriate column vectors using free variables.
1
