Page 286 - 35Linear Algebra
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286 Kernel, Range, Nullity, Rank
16.1 Range
Definition The range of a function f : S → T is the set
ran(f) := {f(s)|s ∈ S} ⊂ T .
It is the subset of the codomain consisting of elements to which the function
f maps, i.e., the things in T which you can get to by starting in S and
applying f.
The range of a matrix is very easy to find; the range of a matrix is the
span of its columns. Thus, calculation of the range of a matrix is very easy
until the last step: simplification. One aught to end by the calculation by
writing the vector space as the span of a linearly independent set.
Example 143 of calculating the range of a matrix.
x x
1 2 0 1 1 2 0 1
y
y
4
ran 1 2 1 2 := 1 2 1 2 ∈ R
|
z
z
0 0 1 1 0 0 1 1
w w
1 2 0 1
2
2
= x + y + z + w x, y, z, w ∈ R .
1
1
0 0 1 1
That is
1 2 0 1 1 2 0 1
,
2
ran 1 2 1 2 = span
,
,
1
2
1
0 0 1 1 0 0 1 1
but since
1 2 0 1 1 2 0 1
RREF 1 2 1 2 = 0 0 1 1
0 0 1 1 0 0 0 0
the second and fourth columns (which are the non-pivot columns), can be expressed
as linear combinations of columns to their left. They can then be removed from the
set in the span to obtain
1 2 0 1 1 0
ran 1 2 1 2 = span .
1
1
,
0 0 1 1 0 1
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