Math 51 TA note class 6
April 22, 2010
1. chap 10 subspaces of Rn
Definition 1. A linear subspace of Rn is a subset V of Rn satisfying the following
properties:
(1) V contains the zero vector.
(2) If v and w are in V then v+w is in V.
(3) If v is in V then cv is in V for any scalar c
Examples of subspaces: N(A), C(A), span {v1 , v2 , . . . , vk }, m-dimensional plane in Rn .
Example 1. Determine whether or not each of the following sets is a subspace. If it is,
show that the three subspace properties are satisfies. If not, show by example that one of
the properties fails.
(1) S=vectors in R3 which lies on the intersection of x+y−z = 0 and 5x+8y−10z =
0.
(2) S= vectors on the line x − 2y = 1 in R2 .
(3) The unit disk x2 + y 2 ≤ 1.
(4) the solution of Ax = 3x, for a given square matrix A.
Example 2. Let V + W = {v + w |v ∈ V and w ∈ W }. Show that if V and W are
subspaces then V + W is a subspace.
Example 3. True/False questions:
(1) Suppose rref(A) contains k pivots, n − k free variables and no 0 = 1 rows, then
N(A) can be written as span of n − k vectors.
(2) There exists a matrix A such that the line 3x − 2y = 1 in R2 equals N(A).
(3) Suppose v1 is the first column of A, then Ax = 2v1 always has a solution.
2
(4) If A is a 2 × 3 matrix, then the
nullspace
is always a subspace in R .
1
0
(5) Let A be a 4 × 2 matrix and
and
are in N(A), then every entry of A is
0
1
zero.
(6) If A is a m × n matrix, and n > m, then the column vectors of A must be linearly
dependent.
2. chap 11 Basis for a subspace
Definition 2. A basis for a subspace V of Rn is a linearly independent set of vectors
{v1 , v2 , . . . , vk } such that V=span{v1 , v2 , . . . , vk }.
Example 4.
(1) R2
3
(2) R
(3) The xy-plane in R3
(4) A line which passes through the origin in R3
1
2
Example 5. Find a basis for the null space of A =
1 1 1 1
.
2 2 3 1
Remark. Suppose A is a m × n matrix. To find a basis for C(A), we can find those
columns in rref(A) which contains a pivot, and the corresponding columns in A will
be a basis for C(A).


1 2 0 ∗ 1
Example 6. Suppose we are given that A = 2 3 ∗ ∗ 4 (* means unknown
0 −1 ∗ 10 2


1 0 0 0 0
number) and rref(A)=0 1 0 0 0. Find a basis of C(A).
0 0 0 0 1
Remark. Let {v1 , v2 , . . . , vk } be a set of vectors in Rn and let A be the matrix which
has column vectors {v1 , v2 , . . . , vk }. Then
(1) rref(A) contains a pivot in each column if and only if {v1 , v2 , . . . , vk } are linearly
inde- pendent.
(2) rref(A) contains a pivot in each row if and only if {v1 , v2 , . . . , vk } spans Rn .
(3) rref(A)=In if and only if k = n and {v1 , v2 , . . . , vk } is a basis for Rn = Rk .
Example
whether the given set S is a basis of the given space.
7. Determine
   
0
2 
 1
(1) 2 , −1 , 0 , R3


1
1
3
2
1
(2)
,
, R2
5
3
1
0
(3)
,
, R2
0
1
Example 8. Find a basis for the following set
1
−1
0
(1) span
,
,
2
1
4
(2) the plane x + y + 2z = 0
3. chap 12 Dimension of a subspace
1 1 0
Example 9. Let A =
, find the rank and nullity of A.
2 1 0