Determine whether or not the given transformation is a linea

Determine whether or not the given transformation is a linear transformation from P_4 to P_4: L(p(x)) = (x + 1)p^n (x + 5) + 2x for p(x) elementof P_4.

Solution

The transformation L has been defined by L: P₄ P₄ such that L(p(x)) = (x+1)p’’(x+5)+2x

Let p₁(x) = a₁x⁴+b₁x³+c₁x²+d₁x +e₁ and p₂(x) = a₂x⁴+b₂x³+c₂x²+d₂x +e₂ be two arbitrary vectors in P₄(x), where a₁, b₁, c₁, d₁, e₁ and a₂, b₂, c₂, d₂, e₂ are arbitrary real numbers and let k be an arbitrary scalar. Then,p₁’(x)= 4a₁x³+3b₁x²+2c₁x+d₁, and p₁’’(x)=12a₁x²+6b₁x+2c₁ so that p₁’’(x+5)=12a₁(x+5)²+6b₁(x +5)+2c₁. Then L(p₁(x))=(x+1)p₁’’(x+5)+2x =(x+1)[12a₁(x+5)²+6b₁(x +5)+2c₁]+2x. Similarly,L(p₂(x))= (x+1)p₂’’(x+5)+ 2x = (x+1)[12a₂(x+5)²+6b₂(x +5)+2c₂]+2x.

Then L(p₁(x)+p₂(x) ) = (x+1)(p₁’’ +p₂’’)(x+5)+2x = (x+1)[12(a₁+a₂)(x+5)²+6(b₁+b₂)(x +5)+2(c₁+c₂)]+2x [(x+1){12a₁(x+5)²+6b₁(x +5)+2c₁}+2x] +[(x+1){12a₂(x+5)²+6b₂(x +5)+2c₂}+2x].

Thus L does not preserve vector addition and hence L is not a linear transformation.