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Posted 3 months ago
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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Answer from Sia
Posted 3 months ago
Solution by Steps
step 1
Consider the set of polynomials P with coefficients in Z5 \mathbb{Z}_5 (the integers modulo 5) and degree less than or equal to 7
step 2
The derivative operator D D sends a polynomial p(x) p(x) to its derivative p(x) p'(x)
step 3
The null space of D D , denoted n n , consists of all polynomials in P P whose derivative is the zero polynomial
step 4
The only polynomials in P P that have a derivative of zero are the constant polynomials. There are 5 constant polynomials in Z5 \mathbb{Z}_5 , one for each element of Z5 \mathbb{Z}_5
step 5
Therefore, the dimension of the null space n n is 1, since the set of constant polynomials forms a basis for the null space
step 6
The range of D D , denoted r r , consists of all possible derivatives of polynomials in P 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 D includes polynomials of degree 0 up to degree 6
step 9
There are 5k 5^k polynomials for each degree k k , from 0 to 6, because each coefficient can be one of 5 elements from Z5 \mathbb{Z}_5
step 10
To find the dimension of the range r r , we sum the number of independent polynomials for each degree: 50+51+52+53+54+55+56 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 1 + 5 + 25 + 125 + 625 + 3125 + 15625 = 19531 . However, since we are working in Z5 \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 r is 7
The dimensions of the null space n n and range r r of D D are 1 and 7, respectively.
Key Concept
Dimension of Null Space and Range of a Linear Operator
The dimension of the null space of the derivative operator on polynomials over Z5 \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).

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