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Linear Algebra

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An Example from Calculus

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If $\mathbb{R}[x]$ is the set of all polynomials in the variable $x$ with real coefficients, then it is a vector space if addition of polynomials is the vector addition and multiplication of a polynomial by a real number is scalar multiplication, and the polynomial $0$ is the zero vector.

As such, the function

$$\cfrac{d}{dx}:\mathbb{R}[x]\rightarrow \mathbb{R}[x]$$

...given by:

$$\cfrac{d}{dx}f(x)=f'(x)$$

...(differentiation) is a linear transformation.

Determine $nullity(\cfrac{d}{dx})$.

A

$0$

B

$1$

C

Infinite

D

It cannot be determined