From Linear Algebra are these transformations linear Are the

From Linear Algebra, are these transformations linear? Are they an isomorphism? Why?

Solution

Yes.The transformation defines a map from P₂ to P₂ .To prove the transformation is linear,the transformation must preserve scalar multiplication,addition and zero vector.

First prove the transformation preserves scalar multiplication

i e,T(f₁+f₂)=T(f₁)+T(f₂) and T(xf₁)=xT(f₁)

T(f₁+f₂)=(f₁₊f₂)(7x)-(f₁₊f₂)(x)

= f₁(7x)-f₁(x)+f₂(7x)-f₂(x)

=T(f₁)+T(f₂)

T(Xf)=XT(f)

T((f₁+f₂)(x))=(f₁+f₂)¹(x)+5x²

=f¹(x)+5x²+f¹₂(x)+5x²=T(f₁)+T(f₂)

To show T is isomorphism ,we need to show tha T is linear transformation and T is invertible .

From above part T is linear,

Now we have to show that T is invertible i.e, T inverse exists

There exists an invertible linear transformation from P₂ to P_2.

Therefore T is isomorfism.