# Modular arithmetic on vector spaces

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1. Feb 23, 2016

### gruba

1. The problem statement, all variables and given/known data
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 $P(x)=U\oplus W$.

2. Relevant equations
-Polynomial vector spaces
-Subspaces
-Modular arithmetic

3. 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?

2. Feb 23, 2016

### 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).

3. Feb 23, 2016

### HallsofIvy

Staff Emeritus
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?