Description
Construct a matrix whose nullspace consists of all combinations of
2
2
1
0
and
3
1
01
.
Problem 2 se column space contains
1
1
5
and
0
3
1
and whose
nullspace contains
1
1
2
.
Problem 3
Let u1 =
10
0
, u2 =
1
1
0
, u3 =
1
1
1
, u4 =
2
3
4
. Show that u1, u2, u3
are independent but u1, u2, u3, u4 are dependent.
1Problem 4 ]
For which numbers c, d does the following matrix have rank 2?
A =
1 2 5 0 5
0 0 c 2 2
0 0 0 d 2
Problem 5
Find a basis for each of the four fundamental subspaces (column, null, row,
left null) associated with the following matrix:
A =
0 1 2 3 4
0 1 2 4 6
0 0 0 1 2
Problem 6
Suppose that S is spanned by s1 =
1
2
2
3
, s2 =
1
3
3
2
. Find two vectors that
span the orthogonal complement S⊥. (Hint: this is the same as solving
Ax = 0 for some A)
Problem 7
Suppose P is the subspace of R4 that consists of vectors
x1
x
2
x
3
x4
that satisfy
x1 + x2 + x3 + x4 = 0. Find a basis for the perpendicular complement P ⊥
of P.
2
