A First Course in Linear Algebra, 2017a

212 R n
Definition 4.106: Image of A
The image of A, written im(A) is given by
im(A)={A⃗x :⃗x ∈ R n }
Consider A as a mapping from R n to R m whose action is given by multiplication. The following
diagram displays this scenario.
null(A)
R n
A →
im(A)
R m
As indicated, im(A) is a subset of R m while null(A) is a subset of R n .
It turns out that the null space and image of A are both subspaces. Consider the following example.
Example 4.107: Null Space
Let A be an m × n matrix. Then the null space of A, null(A) is a subspace of R n .
Solution.
•SinceA⃗0 n =⃗0 m ,⃗0 n ∈ null(A).
•Let⃗x,⃗y ∈ null(A). ThenA⃗x =⃗0 m and A⃗y =⃗0 m ,so
A(⃗x +⃗y)=A⃗x + A⃗y =⃗0 m +⃗0 m =⃗0 m ,
and thus⃗x +⃗y ∈ null(A).
•Let⃗x ∈ null(A) and k ∈ R. ThenA⃗x =⃗0 m ,so
and thus k⃗x ∈ null(A).
A(k⃗x)=k(A⃗x)=k⃗0 m =⃗0 m ,
Therefore by the subspace test, null(A) is a subspace of R n .
♠
The proof that im(A) is a subspace of R m is similar and is left as an exercise to the reader.
We now wish to find a way to describe null(A) for a matrix A. However, finding null(A) is not new!
There is just some new terminology being used, as null(A) is simply the solution to the system A⃗x =⃗0.
Theorem 4.108: Basis of null(A)
Let A be an m × n matrix such that rank(A)=r. Then the system A⃗x =⃗0 m has n − r basic solutions,
providing a basis of null(A) with dim(null(A)) = n − r.
Consider the following example.