linear-guest

16.2 Image 287
It might occur to you that the range of the 3 × 4 matrix from the last
example can be expressed as the range of a 3 × 2 matrix;
⎛ ⎞ ⎛ ⎞
1 2 0 1 1 0
ran ⎝1 2 1 2⎠ = ran ⎝1 1⎠ .
0 0 1 1 0 1
Indeed, because the span of a set of vectors does not change when we replace
the vectors with another set through an invertible process, we can calculate
ranges through strings of equalities of ranges of matrices that differer by
Elementary Column Operations, ECOs, ending with the range of a matrix
in Column Reduced Echelon Form, CREF, with its zero columns deleted.
Example 144 Calculating a range with ECOs
⎛ ⎞ ⎛ ⎞
⎛ ⎞
0 1 1
1 1 0
1 0 0
ran ⎝1 3 1⎠ c 1↔c =
3
ran ⎝1 3 1⎠ c′ 2 =c 2−c 1
= ran ⎝1 2 1
1 2 0
0 2 1
0 2 1
⎛ ⎞ ⎛ ⎞
1 0 0 1 0
c ′ 3 =c 3−c 2
= ran ⎝1 1 0⎠ = ran ⎝1 1⎠ .
0 1 0 0 1
⎠ c′ 2 = 1 2 c 2
⎛
1 0
⎞
0
= ran ⎝1 1 1⎠
0 1 1
This is an efficient way to compute and encode the range of a matrix.
16.2 Image
Definition For any subset U of the domain S of a function f : S → T the
image of U is
f(U) = Im U := {f(x)|x ∈ U} .
Example 145 The image of the cube
⎧ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎫
⎨ 1 0 0
U =
⎩ a ⎬
⎝0⎠ + b ⎝1⎠ + c ⎝0⎠
0 0 1 ∣ a, b, c ∈ [0, 1] ⎭
under multiplication by the matrix
⎛
1 0
⎞
0
M = ⎝1 1 1⎠
0 0 1
287