Page 288 - 35Linear Algebra

288 Kernel, Range, Nullity, Rank

is the parallelepiped

       
1 0 0
 

Img U = a   + b   + c   a, b, c ∈ [0, 1] .
1
1
1

0 0 1
 
Note that for most subsets U of the domain S of a function f the image of
U is not a vector space. The range of a function is the particular case of the
image where the subset of the domain is the entire domain; ranf = ImgS.
For this reason, the range of f is also sometimes called the image of f and is
sometimes denoted im(f) or f(S). We have seen that the range of a matrix
is always a span of vectors, and hence a vector space.
Note that we prefer the phrase “range of f” to the phrase “image of f”
because we wish to avoid confusion between homophones; the word “image”
is also used to describe a single element of the codomain assigned to a single
element of the domain. For example, one might say of the function A : R → R
with rule of correspondence A(x =) = 2x − 1 for all x in R that the image of
2 is 3 with this second meaning of the word “image” in mind. By contrast,
one would never say that the range of 2 is 3 since the former is not a function
and the latter is not a set.
For thinking about inverses of function we want to think in the oposite
direction in a sense.
Definition The pre-image of any subset U ⊂ T is

f −1 (U) := {s ∈ S|f(s) ∈ U} ⊂ S.

The pre-image of a set U is the set of all elements of S which map to U.
   
2
 

Example 146 The pre-image of the set U = a   a ∈ [0, 1] (a line segment)
1

 
1
under the matrix
 
1 0 1
3
M =  0 1 1  : R → R 3
0 1 1
is the set
M −1 U = {x|Mx = v for some v ∈ U}

      
 x 1 0 1 x 2

1
=   0 1 1    = a   for some a ∈ [0, 1] .
y
y

z 0 1 1 z 1
 
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