Modular arithmetic on vector spaces

In summary, U is the subspace of P(x) that includes all polynomials of degree n- 2 that are not equal to 0 or 3.
  • #1
gruba
206
1

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?
 
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  • #2
gruba said:

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]
How does modular arithmetic come into this problem? In the problem statement, you say only that u(3) = 0 and u(-2) = 0.
gruba said:
[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]
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).
gruba said:
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?
 
  • #3
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?
 

1. What is modular arithmetic on vector spaces?

Modular arithmetic on vector spaces is a mathematical concept that involves performing arithmetic operations on vectors using modular arithmetic. This means that instead of using traditional addition and multiplication, the operations are performed within a specific modulus, or a fixed number. This allows for easier computation and can be useful in various applications.

2. How is modular arithmetic used in vector spaces?

Modular arithmetic is used in vector spaces to simplify computations and to make it easier to work with large numbers. It can also be used to find patterns and relationships between vectors. Additionally, it is commonly used in coding theory and cryptography to encrypt and decrypt messages.

3. What are some real-world applications of modular arithmetic on vector spaces?

Modular arithmetic on vector spaces has many real-world applications, including in computer graphics, error-correcting codes, and cryptography. It is also used in the design of communication networks and in data compression algorithms.

4. Can modular arithmetic on vector spaces be used in higher dimensions?

Yes, modular arithmetic on vector spaces can be extended to higher dimensions. The principles and operations remain the same, but the calculations become more complex as the number of dimensions increases.

5. Are there any limitations to using modular arithmetic on vector spaces?

One limitation of using modular arithmetic on vector spaces is that it only works with integer values. This means that fractions or decimals cannot be used. Additionally, the modulus must be a positive integer, and the operations must be performed within the same modulus for accurate results.

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