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2.    f: C([0,1]) -» K    fig) = J₍₎¹ g{x)dx jest liniowe

/(«5i + P9i) = Jo («5i + P92)(.x)dx = a g₂ (x)dx + fi f₀' g₂ (x)dx = af(g_t) + Pf(g₂)

3.    /':R²-»R f(x,y) - x y + 1 nie jest liniowe /^r(a(x¹,y¹)+/?(x²,y²))=/'((ax¹ +/?x²,ay¹ + /?y²)) = (a*¹ + (3x²)(ay^l + /?y²) + 1 =

= a²x¹y¹ + a/?(x¹y² + x²y¹) + (3²x²y² + 1 a/(*V) + Pf(x²y²) = ax¹y¹ + a + px²y² + /?

4.    f: IR³ -»IR², f(x¹,x²,x³)=(2x¹ - x², x³ + 3x^Ł) jest liniowe a,PelR, a,b elR³

f(aa+ pb)= f(a(a¹,a²,a³)+ P(b¹,b²,b³))= f((aa³+ Pb¹, aa²+ Pb², aa³+ Pb³))=

(2- (aa¹-!- pb³)-( aa²+ pb²), (aa³+ Pb^+Sjaa¹-!- pb¹)=(2aa¹- aa²+2 Pb¹- Pb², aa³+3aa¹+ pb³+3pb¹)= (2aa¹- aa², aa³+3aa¹)+(2 pb¹- Pb², pb³+3Pb¹)= a(2a¹-a²,a³+3a¹)+ P(2b¹-b²,b³+3b¹)= a f(a¹,a²,a³)+ p f(b¹,b²,b³)= a f(a)+ p f(b)

5.    f:IR³ -»IR², f(x¹,x²,x³)=( X¹ • x², x³ - 3x¹) nie jest liniowe a,pelR, a,b elR³

f (aa+ pb)= f (aa'+ pb¹, aa²+ pb², aa³+ pb³)=(( aa¹-!- pb¹)! aa²+ pb²), aa³+ pb³-l)=

( a²a¹a²+ aP(a¹b²+b¹a²)+ P²b¹b², aa³+ pb³-l)

a f(a)+ P f(b)=( aa³a², aa³ -a)+( P^b², Pb³- P)=( aa^²-!- P^b², aa³+ Pb³- a-P)

L*P - odwzorowanie nie jest liniowe

6.    f:IR₂[x] —*IRIx] f(w)=w' jest liniowe a,P elR w,velR₂[x]

f (aw+ pv)=( aw+ pv) '=( aw)'+( pv) '= aw'+ pv'=af(w)+ p f(v)