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MTH 309 30. Linear transformations of vector spaces
DefinitionLet V � W be vector spaces A linear transformation is a function
T : V → W
which satisfies the following conditions:1) T (u + v) = T (u) + T (v) for all u� v ∈ V2) T (�v) = �T (v) for any v ∈ V and any scalar �.
153
Example : If A is an mxn matrix then
it defines a linear transformation :
Tai Tan→Rm
✓1-Av
ExampleiRecall : c- (IR) = { the vector space of
all
smooth functions f : IR→IR}Take D : C
-CR)→ c- (IR) , D ( f) = f'←the derivativeof f .
D is a linear transformation-D D (ft g) = #tf)
' = f ' + g' = Dcf) t Dlg)
2) D ( cf) = @f)'= c. f
'= c.Dlf) .
Note. If T : V → W is a linear transformation then for any vector b ∈ W wecan consider the equation
T (x) = b
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Example :-
If Ta : Rn→Rm - a (matrix) linear transformation✓ 1-7 Av
then the equation TAG) - b is the same as the
matrix equation Ax - b.
Example :-
Take D: CIR)→ (R)
f- It)- f' ft)
For GE C- (IR) the equation D Cx) - g
is the same as the differential equationDXIt
= 8
This equation is solved by integration :
xCt) = JgHolt
DefinitionIf T : V → W is a linear transformation then:1) The kernel of T is the set
Ker(T ) = {v ∈ V | T (v) = 0}
2) The image of T is the set
Im(T ) = {w ∈ W | w = T (v) for some v ∈ V }
155
I*#IEt⇒ExampleiA - mxn matrix Ta Rn→ Rm
V 1-2 Av
Ker(Ta) = { vets" I Ta (v) - O }= f v EIR " I Av -- O } = Nul CA)
TthenuH spaceof A
Tm (TA) = { b ERMITA(v) - b for some ve IR" }= { b. ERM I Av - b - i.- "- }= Col (A) ← the column space of A .
Exampte :D , cocky → C- UR)
f- 1→ f"
the set of all
Ker (D) = {FE (IR) I f'-Of. = { constant functions }
Im (D) = Ige (R) I g -- f'for some f-ECHR) } = c- (IR)
Proposition
If T : V → W is a linear transformation then:1) Ker(T ) is a subspace of V2) Im(T ) is a subspace of W
TheoremIf T : V → W is a linear transformation and �0 is a solution of the equation
T (x) = b
then all other solutions of this equation are vectors of the form
v = v0 + n
where n ∈ Ker(T ).
156
ProofIf is a solution of TX) -- b and ne Ker CT)then
Thoth) =Tho)TT (n) = b t@ = b
sci Voth is also a solution of T(x) -- b .
Proof of the converse issimilar .
Example.
D : C∞(R) −→ C∞(R)
� �−→ � �
157
RiccaKer (D) = { all constant functions}
Let get) -- E
solutions of D = g are functions fsuch that f
' ft) = glt) =t2
these solutions of D (x) - g
are functions
f- Ct) -- Stolt = It't C- -
T Ta particular a constant functionsolution of i.e a functionD. 67--8 from Ker (D) .