Question

Let denote the vector space of polynomials in the variable of degree or less with real coefficients. let be the function that sends a polynomial to its derivative. that is, for all polynomials . is a linear transformation?

A) Yes, Yes, Yes
B) Yes, No, No
C) No, Yes, No
D) No, No, Yes

Answer 1

The function mapping a polynomial to its derivative satisfies both properties of additivity and scalar multiplication, making it a linear transformation.

Step-by-step explanation:

The question is asking whether the given function, which maps a polynomial to its derivative, is a linear transformation. To determine if a function is a linear transformation, the function must satisfy two properties: additivity and scalar multiplication.

To check additivity, let's assume f and g are two polynomials from the vector space. The derivative of the sum f+g is f'(x) + g'(x), which means the function distributes over addition. To check scalar multiplication, let c be a scalar, then the derivative of cf is c * f'(x), which shows the function respects scalar multiplication.

Since both conditions for a linear transformation are met, the correct choice is Yes, the function is a linear transformation. Thus, the answer to the student's question would be option A) Yes, Yes, Yes.

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