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Modular arithmetic on vector spaces

  1. Feb 23, 2016 #1
    1. The problem statement, all variables and given/known data
    Let [itex]U[/itex] is the set of all polynomials [itex]u[/itex] on field [itex]\mathbb F[/itex] such that [itex]u(3)=u(-2)=0[/itex]. Check if [itex]U[/itex] is the subspace of the set of all polynomials [itex]P(x)[/itex] on [itex]\mathbb F[/itex] and if it is, determine the set [itex]W[/itex] such that [itex]P(x)=U\oplus W[/itex].

    2. Relevant equations
    -Polynomial vector spaces
    -Modular arithmetic

    3. The attempt at a solution
    [itex]U=\{u(x):u(x)\mod (x-3)=0 \land u(x)\mod (x+2)= 0\}[/itex]

    [itex]U[/itex] is the subspace of [itex]P(x)[/itex] iff

    [itex]1)[/itex] [itex]\forall u_1,u_2\in U\Rightarrow u_1+u_2\in U[/itex]

    [itex]2)[/itex] [itex]\forall u\in U,\forall \alpha\in \mathbb F\Rightarrow \alpha u\in U[/itex]

    How to check if [itex]U[/itex] is the subspace of [itex]P(x)[/itex]?

    Assuming [itex]U[/itex] is the subspace of [itex]P(x)\Rightarrow[/itex]

    [itex]P(x)=U\oplus W=U+W \mod n[/itex]

    where [itex]n[/itex] should be the total number of polynomials in [itex]U[/itex] and [itex]W[/itex].

    This means that [itex]W[/itex] is the set of all polynomials [itex]u(x)[/itex] defined as
    [itex]W=\{u(x): u(3)\neq u(-2)\neq 0 \lor u(3)\neq u(-2)=0\lor u(3)=u(-2)\neq 0\}[/itex]

    Is this correct?
  2. jcsd
  3. Feb 23, 2016 #2


    Staff: Mentor

    How does modular arithmetic come into this problem? In the problem statement, you say only that u(3) = 0 and u(-2) = 0.
    Minor point, but ##\Rightarrow## is not appropriate in the above. It should be used when one statement implies another statement.##\forall u \in U## is not a statement (i.e., a sentence whose truth value can be determined).
  4. Feb 23, 2016 #3


    User Avatar
    Science Advisor

    Any polynomial, u(x), of degree n, such that u(3)= u(-2)= 0 is of the form (x- 3)(x+ 2)v(x) where v is a polynomial of degree n- 2. To show this is a subspace of the space of all polynomials, you only need to show that this set is closed under addition and scalar multiplication- If u(x)= (x- 3)(x+ 2)v(x) and w(x)= (x- 3)(x+ 2)y(x), what can you say about u+ w? What can you say about au where a is a number?
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