Problem 1
Construct a matrix whose nullspace consists of all combinations of 2 and
3 1.
0 1
Problem 2 [15 pts]
Construct a matrix whose column space contains 1 and 3 and whose
1 nullspace contains 1.
2
Problem 3 [10 pts] 1
1 1 2
Let u1 = 0, u2 = 1, u3 = 1, u4 = 3. Show that u1,u2,u3 0014
are independent but u1, u2, u3, u4 are dependent.
1
1 0 51
2 1
0
Problem 4 [15 pts]
For which numbers c, d does the following matrix have rank 2?
12505 A=00c22
000d2
Problem 5 [15 pts]
Find a basis for each of the four fundamental subspaces (column, null, row,
left null) associated with the following matrix:
01234 A=01246
00012
Problem 6 [15 pts]
Suppose that S is spanned by s1 = 2, s2 = 3. Find two vectors that
1 1 2 3
32
span the orthogonal complement S⊥. (Hint: this is the same as solving Ax = 0 for some A)
Problem 7 [15 pts]
Suppose P is the subspace of R4 that consists of vectors x2 that satisfy
x4
x1 + x2 + x3 + x4 = 0. Find a basis for the perpendicular complement P ⊥
of P.
2
x1 x 3
