Linear Algebra, 2020a

174 Chapter Three. Maps Between Spaces
1.1 Example The space of two-wide row vectors and the space of two-tall column
vectors are “the same” in that if we associate the vectors that have the same
components, e.g.,
( )
1
(1 2) ←→
2
(read the double arrow as “corresponds to”) then this association respects the
operations. For instance these corresponding vectors add to corresponding totals
( ) ( ) ( )
1 3 4
(1 2)+(3 4)=(4 6) ←→ + =
2 4 6
and here is an example of the correspondence respecting scalar multiplication.
( ) ( )
1 5
5 · (1 2)=(5 10) ←→ 5 · =
2 10
Stated generally, under the correspondence
(a 0 a 1 ) ←→
( )
a 0
a 1
both operations are preserved:
(a 0 a 1 )+(b 0 b 1 )=(a 0 + b 0 a 1 + b 1 ) ←→
( ) ( ) ( )
a 0 b 0 a 0 + b 0
+ =
a 1 b 1 a 1 + b 1
and
r · (a 0 a 1 )=(ra 0 ra 1 ) ←→ r ·
( ) ( )
a 0 ra 0
=
a 1 ra 1
(all of the variables are scalars).
1.2 Example Another two spaces that we can think of as “the same” are P 2 , the
space of quadratic polynomials, and R 3 . A natural correspondence is this.
⎛
a 0 + a 1 x + a 2 x 2 ⎜
a ⎞
⎛
0
⎟
←→ ⎝a 1 ⎠ (e.g., 1 + 2x + 3x 2 ⎜
1
⎞
⎟
←→ ⎝2⎠)
a 2 3
This preserves structure: corresponding elements add in a corresponding way
⎛
a 0 + a 1 x + a 2 x 2
+ b 0 + b 1 x + b 2 x 2 ⎜
a ⎞ ⎛
0
⎟ ⎜
b ⎞ ⎛
0
⎟ ⎜
a 0 + b ⎞
0
⎟
←→ ⎝a 1 ⎠ + ⎝b 1 ⎠ = ⎝a 1 + b 1 ⎠
(a 0 + b 0 )+(a 1 + b 1 )x +(a 2 + b 2 )x 2 a 2 b 2 a 2 + b 2