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gruba
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Homework Statement
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].
Homework Equations
-Polynomial vector spaces
-Subspaces
-Modular arithmetic
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?