Download Linear Transformations

LINEAR TRANSFORMATIONS
ROHAN RAMCHAND
Let V, W be two vector spaces over a field F .
Definition 1. A linear transformation is a mapping T : V → W such that for u, v ∈ V
and λ ∈ F ,
T (u + v) = T (u) + T (v)
T (0) = 0
T (−u) = −T (u)
T (λu) = λT (u)
Let T : F n → F m be a linear transformation from an n-dimensional field to itself. Then
T can be represented as an m-dimensional vector of linear transformations from F n to
F:
T = (T1 , T2 , ..., Tm ), Ti : F n → F.
Further, each Ti is a linear transformation:
Ti (x1 , x2 , ..., xn ) = ai1 x1 + ai2 x2 + ... + ain xn .
Then any linear transformation can be represented as an n × m matrix:



T =

a11
a21
..
.
a12
a22
..
.
a1n
a2n
..
.
...
...
..
.
am
am
. . . am
n
1
2





Example. Let T (x, y) = (x + y, x − y) be a linear transformation from R2 → R2 . Then
T =
1 1
1 −1
.
Example. Let X, Y be finite sets and ϕ : X → Y be a map from X to Y . Let F (X), F (Y )
be the field of F -valued functions on X and Y , respectively, and let ϕ∗ : F (Y ) → F (X) be
a mapping of fields such that ϕ∗ (f ) = f ◦ ϕ. Then ϕ∗ is a linear transformation.
1
LINEAR TRANSFORMATIONS
2
Proof.
ϕ∗ (f + g)(x) = (f + g)(ϕ(x))
= f (ϕ(x)) + g(ϕ(x))
= ϕ∗ (f )(x) + ϕ∗ (g)(x)
ϕ∗ (λf ) = (λf )(ϕ(x))
= λf (ϕ(x))
= λϕ∗ (f )(x)
Therefore, ϕ∗ satisfies both properties of a linear transformation and the proof is complete.
Operations on Linear Transformations
We define two operations on linear transformations, addition and multiplication by a
scalar.
Definition 2. Let T, S : V → W be linear transformations from a vector space V to a
vector space W over the field F . Then
(T + S)(v) = T (v) + S(v).
Definition 3. Let T : V → W be a linear transformation from a vector space V to a vector
space W over the field F . Then for some λ ∈ F ,
(λ · T )(v) = λ · T (v).
The following theorem is presented without proof.
Theorem 1. Let T, S : V → W and λ ∈ F as above. Then both T + S and λ · T are linear
transformations.
Definition 4. Let V and W be two vector spaces over a field F . Then Hom(V, W ) is the
set of linear transformations from V to W .
Theorem 2. Let + : Hom(V, W ) × Hom(V, W ) → Hom(V, W ) and α : F × Hom(V, W ) →
Hom(V, W ) the operations on linear transformations as defined above. Then the triple
(Hom(V, W ), +, α) is a vector space over F .
(Hom(V, W ) is also referred to as the tensor product of V and W -dual, V
L
W + .)
Proof. (Hom(V, W ), +) is an abelian group by definition. Let 0 : V → W be the identity
function on Hom(V, W ); then 0V = 0W ∀ v ∈ V . Finally, define the additive inverse as
(−T )(v) = T (−v) = −T (v). Then (Hom(V, W ), +, α) is a vector space.
LINEAR TRANSFORMATIONS
3
Note that Hom(Hom(V, W ), V ) is a vector space, as is Hom(Hom(Hom(V, W ), V ), W ), etc.
This is known as the self-reducibility of linear algebra.
 1

a1 . . . a1n


Let T, S : F n → F m be linear transformations. Then, as defined above, T ↔  ... . . . ... 
am
. . . am
n
1
 1



b1 . . . b1n
c11 . . . c1n




and S ↔  ... . . . ... . Then T + S ↔  ... . . . ... , where cji = aji + bji .
bm
. . . bm
cm
. . . cm
n
n
1
1
Definition 5. Let Matm×n (F ) be the set of all m × n matrices with elements in F . Let
+ : Mat(F ) × Mat(F ) → Mat(F ) and α : F × Mat(F ) → Mat(F ); (Matm×n (F ), +, α) is a
vector space.
With this definition, the following theorem is presented without proof.
Theorem 3. Let Hom(F n , F m ) be the set of linear transformations from F n to F m , and
let Matm×n (F ) be the set of m × n matrices with elements in F . Then Hom(F n , F m ) ∼
=
Matm×n (F ).
We now define one final operation on linear transformations, composition of transformations.
Definition 6. Let U, V, W be three vector spaces over a field F . Let T : U → V and
S : V → W be linear transformations. Then we define composition of linear transformations as S ◦ T : U → W = S(T (u)), u ∈ U .
Theorem 4. Let S ◦ T be the composition of linear transformations S and T . Then S ◦ T
is a linear transformation.
Proof.
(S ◦ T )(u1 + u2 ) = S(T (u1 + u2 ))
= S(T (u1 ) + T (u2 ))
= S(T (u1 )) + S(T (u2 ))
(S ◦ T )(λ · u) = S(T (λ · u))
= S(λ · T (u))
= λ · S(T (u))
Then S ◦ T is a linear transformation and the proof is complete.
LINEAR TRANSFORMATIONS
4
n
n
Finally,
 as1 above, 1letT, S : F →
 F 1 be linear
transformations. Then,
 1 as defined
 above,
a1 . . . a n
b1 . . . b1n
c1 . . . c1n






T ↔  ... . . . ...  and S ↔  ... . . . ... . Then S ◦ T ↔  ... . . . ... , where
an1 . . . ann
bn1 . . . bnn
cn1 . . . cnn
cji = b1i aj1 +b2i aj2 +...+bni ajn . Therefore, S ◦T = T S. (This is presented without proof.)