Suppose P is the set of polynomials with coefficients in Z_5 and degree less than or equal to 7. If the operator D sends p(x) in P to its derivative p′(x), what are the dimensions of the null space n and range r of D?

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∻Solution by Steps∻ ‖ step 1 ⋮ Consider the set of polynomials P with coefficients in $ \mathbb{Z}_5 $ (the integers modulo 5) and degree less than or equal to 7. ‖ ‖ step 2 ⋮ The derivative operator $ D $ sends a polynomial $ p(x) $ to its derivative $ p'(x) $. ‖ ‖ step 3 ⋮ The null space of $ D $, denoted $ n $, consists of all polynomials in $ P $ whose derivative is the zero polynomial. ‖ ‖ step 4 ⋮ The only polynomials in $ P $ that have a derivative of zero are the constant polynomials. There are 5 constant polynomials in $ \mathbb{Z}_5 $, one for each element of $ \mathbb{Z}_5 $. ‖ ‖ step 5 ⋮ Therefore, the dimension of the null space $ n $ is 1, since the set of constant polynomials forms a basis for the null space. ‖ ‖ step 6 ⋮ The range of $ D $, denoted $ r $, consists of all possible derivatives of polynomials in $ P $. ‖ ‖ step 7 ⋮ A polynomial of degree 7 has a derivative of degree 6, a polynomial of degree 6 has a derivative of degree 5, and so on, down to a linear polynomial, which has a constant derivative. ‖ ‖ step 8 ⋮ Therefore, the range of $ D $ includes polynomials of degree 0 up to degree 6. ‖ ‖ step 9 ⋮ There are $ 5^k $ polynomials for each degree $ k $, from 0 to 6, because each coefficient can be one of 5 elements from $ \mathbb{Z}_5 $. ‖ ‖ step 10 ⋮ To find the dimension of the range $ r $, we sum the number of independent polynomials for each degree: $ 5^0 + 5^1 + 5^2 + 5^3 + 5^4 + 5^5 + 5^6 $. ‖ ‖ step 11 ⋮ Simplifying, we get $ 1 + 5 + 25 + 125 + 625 + 3125 + 15625 = 19531 $. However, since we are working in $ \mathbb{Z}_5 $, we only need to count the number of degrees, which is 7 (from 0 to 6). ‖ ‖ step 12 ⋮ Therefore, the dimension of the range $ r $ is 7. ‖ ∻Answer∻ ⚹ The dimensions of the null space $ n $ and range $ r $ of $ D $ are 1 and 7, respectively. ⚹ ∻Key Concept∻ ⚹Dimension of Null Space and Range of a Linear Operator⚹ ∻Explanation∻ ⚹The dimension of the null space of the derivative operator on polynomials over $ \mathbb{Z}_5 $ with degree at most 7 is the number of constant polynomials, which is 1. The dimension of the range is the number of possible degrees of the derivatives, which is 7 (from degree 0 to degree 6).⚹◊What is the rank-nullity theorem and how is it related to the dimensions of the null space and range?⍭ Generate me a similar question◊

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