Section 4.2
Null Spaces, Column Spaces, and Linear Transformations
In this section we will look at two
important subspaces associated with a
matrix, the null space and the column
space.
Consider the following system of homogeneous
equations:
x1  3x2  2 x3  0
(1)
 5 x1  9 x2  x3  0
Recall that in matrix
form, this system may be


written as Ax  0
where
 1  3  2
A : 
 5

x
9
1 
Remember that the set of all
that satisfy (1) is called
the solution set of the system. Often it is convenient to
relate this solution
set directly to the matrix A and the
 
equation Ax  0 .
Definition
The null space of an m n matrix A, written as Nul A, is
the set of all solutions to Ax=0.

Nul A  x | Ax  0 and x  Rn

 1  3  2
Example: Find the null space of 


5
9
1


Theorem 2
The null space of an m n matrix A is a subspace of  n .
Proof:
 
Solving the equation Ax  0 produces an explicit
description of Nul A.
Ex: Find a spanning set for the null space of
1 4  3  3 1 
0 2  4 1 0 


0 0 0
0 0
For all problems of the previous type, the following is
always true:
• The spanning set (produced using the method shown) is
automatically linearly independent because the free
variables are the weights of the spanning vectors.
•When Nul A contains the nonzero vectors, the number of
vectors in the spanning set for Nul A equals the number of
free variables in the equation Ax=0.
Definition
The column space of an m n matrix A, written as Col A, is
the set of all linear combinations of the columns of A.
Col A  Span{a1 ,, an }
Note: Col A  {b | Ax  b for some x  n }
Theorem 3
m

The column space of an m n matrix A is a subspace of
.
2a  3b

Example:


Find a matrix A such that W= Col A, where W   a  b  : a, b  


  3b 




Recall from Theorem 4 in Section 1.4 that the
columns of A span Rm if and only if the equation
___________ has a solution for
_______________________
The column space of an ________ matrix A is all of
______ if and only if the equation __________ has a
solution for each ____ in ______.
4  2 1
2
Let A   2  5 7 3
 3
7  8 6
Example:
1. If the column space of A is a subspace of  k , what is k?
2. If the null space of A is a subspace of  k , what is k?
3. Find a nonzero vector in Col A and a nonzero vector in Nul A.
 3  3
4. Determine if   2  ,   1 are in Nul A and in Col A.
   
  1   3 
 
0 
Definition:
A linear transformation T from a vector space V into a vector
space W is a rule assigns to each vector x in V a unique vector
T(x) in W, such that
(i) T(u+v)=T(u)+T(v) for all u, v in the domain of T:
(ii) T(cu)=cT(u) for all u and all scalars c.
kernelof T  x | T ( x)  0 and x V 
Range of T  {b W | T ( x)  b for some x V }
If T(x)=Ax for some matrix A,
Kernel of T = Nul A
Range of T = Col A.