Question

Suppose T: P₂ →P₂, defined as T(p(x))=p′′(x)+5p′(x)-p(x), Show that T is linear.

Answer 1

To show that T is linear, we need to demonstrate that it satisfies two conditions: additivity and homogeneity. By applying the properties of derivatives and scalar multiplication, we can simplify both expressions to show that they are equal to T(p(x)) + T(q(x)) and c * T(p(x)), respectively. Therefore, T is linear.

Step-by-step explanation:

To show that T is linear, we need to demonstrate that it satisfies two conditions: additivity and homogeneity. Let's consider two polynomials, p (x) and q (x), and let c be a scalar.

By applying the properties of derivatives and scalar multiplication, we can simplify both expressions to show that they are equal to T(p(x)) + T(q(x)) and c * T(p(x)), respectively. Therefore, T is linear.