Question

Math

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 $\mathbb{Z}_5$ (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).

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