Question

1   Define by T:P2-P2 is given by T(p(x))=p(x)-p'(x)
a. Prove that T is a linear transformation.
b. Show T is one to one.
c. If   is given by . Explain why T is not one to one.

Answer 1

Part - (a)

Let p(x), q(x) are in P₂ . Let c is in the scaler field.

T(cp(x) + q(x))

= cp(x) + q(x) - (cp(x) + q(x))'

= cp(x) + q(x) - cp'(x) - q'(x)

= c(p(x) - p'(x)) + (q(x) - q'(x))

= cT(p(x)) + T(q(x))

Therefore T is a linear transformation.

Part - (b)

Let T(p(x)) = T(q(x))

=> p(x) - p'(x) = q(x) - q'(x)

=> p(x) - q(x) = p'(x) - q'(x)

Now, the LHS is a polynomial of degree 2 , and RHS is a polynomial of degree 1 . So, the coefficients of highest degree of p(x) - q(x) must be 0 . That is, the coefficients of highest degree of p(x) and q(x) must agree . Then can equate the coefficients . Simply, let p(x) = a₀x² + a₁x + a₂ and q(x) = b₀x + b₁x + b₂

Equating the coefficients of the powers of x , we get

p(x) = q(x)

so , T is one - one .

Part - (c)

The question is not complete . Some parts are missing in the question.