Linear Algebra, 2020a

176 Chapter Three. Maps Between Spaces
To prove that f is onto we must check that any member of the codomain R 2
is the image of some member of the domain G. So, consider a member of the
codomain ( )
x
y
and note that it is the image under f of x cos θ + y sin θ.
Next we will verify condition (2), that f preserves structure. This computation
shows that f preserves addition.
f ( (a 1 cos θ + a 2 sin θ)+(b 1 cos θ + b 2 sin θ) )
= f ( (a 1 + b 1 ) cos θ +(a 2 + b 2 ) sin θ )
( )
a 1 + b 1
=
a 2 + b 2
( ) ( )
a 1 b 1
= +
a 2 b 2
= f(a 1 cos θ + a 2 sin θ)+f(b 1 cos θ + b 2 sin θ)
The computation showing that f preserves scalar multiplication is similar.
f ( r · (a 1 cos θ + a 2 sin θ) ) = f( ra 1 cos θ + ra 2 sin θ )
( )
ra 1
=
ra 2
( )
a 1
= r ·
a 2
= r · f(a 1 cos θ + a 2 sin θ)
With both (1) and (2) verified, we know that f is an isomorphism and we
can say that the spaces are isomorphic G = ∼ R 2 .
1.5 Example Let V be the space {c 1 x + c 2 y + c 3 z | c 1 ,c 2 ,c 3 ∈ R} of linear combinations
of the three variables under the natural addition and scalar multiplication
operations. Then V is isomorphic to P 2 , the space of quadratic polynomials.
To show this we must produce an isomorphism map. There is more than one
possibility; for instance, here are four to choose among.
c 1 x + c 2 y + c 3 z
f
↦−→
1
c1 + c 2 x + c 3 x 2
f
↦−→
2
c2 + c 3 x + c 1 x 2
f
↦−→
3
−c1 − c 2 x − c 3 x 2
f
↦−→
4
c1 +(c 1 + c 2 )x +(c 1 + c 3 )x 2