Question

Let Pₙ denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: P₃ → P₂ be the function that sends a polynomial to its derivative. That is, D(p(x))= p′(x) for all polynomials p(x) ∈ P₃ p(x) ∈ P₃. Is D a linear transformation?

Answer 1

The function D, which maps a polynomial to its derivative, is a linear transformation because it satisfies both additivity and homogeneity, the two main properties required for a function to be considered linear.

Step-by-step explanation:

The question asks whether the derivative operator D, defined as D(p(x))= p′(x) for all polynomials p(x) in the vector space P3 of polynomials of degree 3 or less with real coefficients, is a linear transformation when it maps to the vector space P2 of polynomials of degree 2 or less. For D to be considered a linear transformation, it needs to satisfy two properties: additivity and homogeneity.

To demonstrate the additivity property, consider two polynomials f(x) and g(x) in P3. According to the additivity condition for linear transformations, D(f(x) + g(x)) should be equal to D(f(x)) + D(g(x)). Since the derivative of a sum is the sum of the derivatives, this property holds true for D.

To demonstrate homogeneity, consider a polynomial f(x) in P3 and a scalar c in the field of real numbers. The homogeneity condition requires that D(c*f(x)) = c*D(f(x)). As differentiation is consistent with scalar multiplication, the derivative of a scalar multiple of a function is the scalar multiple of the derivative of the function, and hence this property is also satisfied.

Since both additivity and homogeneity hold, the function D is indeed a linear transformation.