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 �.

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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 }

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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 ).

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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)

� �−→ � �

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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) .