Modular arithmetic on vector spaces

[gruba]

Homework Statement

Let U is the set of all polynomials u on field \mathbb F such that u(3)=u(-2)=0. Check if U is the subspace of the set of all polynomials P(x) on \mathbb F and if it is, determine the set W such that P(x)=U\oplus W.

Homework Equations

-Polynomial vector spaces
-Subspaces
-Modular arithmetic

The Attempt at a Solution

U=\{u(x):u(x)\mod (x-3)=0 \land u(x)\mod (x+2)= 0\}

U is the subspace of P(x) iff

1) \forall u_1,u_2\in U\Rightarrow u_1+u_2\in U

2) \forall u\in U,\forall \alpha\in \mathbb F\Rightarrow \alpha u\in U

How to check if U is the subspace of P(x)?

Assuming U is the subspace of P(x)\Rightarrow

P(x)=U\oplus W=U+W \mod n

where n should be the total number of polynomials in U and W.

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

Is this correct?

 

[Mark44]

Homework Statement

Let U is the set of all polynomials u on field \mathbb F such that u(3)=u(-2)=0. Check if U is the subspace of the set of all polynomials P(x) on \mathbb F and if it is, determine the set W such that P(x)=U\oplus W.

Homework Equations

-Polynomial vector spaces
-Subspaces
-Modular arithmetic

The Attempt at a Solution

U=\{u(x):u(x)\mod (x-3)=0 \land u(x)\mod (x+2)= 0\}

How does modular arithmetic come into this problem? In the problem statement, you say only that u(3) = 0 and u(-2) = 0.

U is the subspace of P(x) iff

1) \forall u_1,u_2\in U\Rightarrow u_1+u_2\in U

2) \forall u\in U,\forall \alpha\in \mathbb F\Rightarrow \alpha u\in U

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).

How to check if U is the subspace of P(x)?

Assuming U is the subspace of P(x)\Rightarrow

P(x)=U\oplus W=U+W \mod n

where n should be the total number of polynomials in U and W.

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

Is this correct?

 

[HallsofIvy]

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?