Linear Algebra, 2020a

196 Chapter Three. Maps Between Spaces
operations. Let f, g: V → W be linear. Then the operation of function addition
is preserved
(f + g)(c 1 ⃗v 1 + c 2 ⃗v 2 )=f(c 1 ⃗v 1 + c 2 ⃗v 2 )+g(c 1 ⃗v 1 + c 2 ⃗v 2 )
= c 1 f(⃗v 1 )+c 2 f(⃗v 2 )+c 1 g(⃗v 1 )+c 2 g(⃗v 2 )
( ) ( )
= c 1 f + g (⃗v1 )+c 2 f + g (⃗v2 )
as is the operation of scalar multiplication of a function.
Hence L(V, W) is a subspace.
(r · f)(c 1 ⃗v 1 + c 2 ⃗v 2 )=r(c 1 f(⃗v 1 )+c 2 f(⃗v 2 ))
= c 1 (r · f)(⃗v 1 )+c 2 (r · f)(⃗v 2 )
QED
We started this section by defining ‘homomorphism’ as a generalization of
‘isomorphism’, by isolating the structure preservation property. Some of the
points about isomorphisms carried over unchanged, while we adapted others.
Note, however, that the idea of ‘homomorphism’ is in no way somehow
secondary to that of ‘isomorphism’. In the rest of this chapter we shall work
mostly with homomorphisms. This is partly because any statement made about
homomorphisms is automatically true about isomorphisms but more because,
while the isomorphism concept is more natural, our experience will show that
the homomorphism concept is more fruitful and more central to progress.
Exercises
̌ 1.18 Decide ⎛ if ⎞each h: R 3 → R 2 is linear. ⎛ ⎞
⎛ ⎞
x ( )
x ( x (
(a) h( ⎝
x
y⎠) =
(b) h( ⎝ 0
y⎠) = (c) h( ⎝ 1
y⎠) =
x + y + z
0)
1)
z
z
z
⎛ ⎞
x ( )
(d) h( ⎝ 2x + y
y⎠) =
3y − 4z
z
̌ 1.19 Decide if each map h: M 2×2 → R is linear.
( ) a b
(a) h( )=a + d
c d
( ) a b
(b) h( )=ad − bc
c d
( ) a b
(c) h( )=2a + 3b + c − d
c d
( ) a b
(d) h( )=a 2 + b 2
c d
̌ 1.20 Show that these are homomorphisms. Are they inverse to each other?
(a) d/dx: P 3 → P 2 given by a 0 + a 1 x + a 2 x 2 + a 3 x 3 maps to a 1 + 2a 2 x + 3a 3 x 2
(b) ∫ : P 2 → P 3 given by b 0 + b 1 x + b 2 x 2 maps to b 0 x +(b 1 /2)x 2 +(b 2 /3)x 3