Linear Algebra - Polynomial Subsets of Subspaces

In summary: Is this set a subspace of P2?b) The set of all vectors of the form a2t2 + a1t1 + a0 (where a2, a1, and a0 are scalar-valued dummy variables) for which a1 = 2a0. Is this set a subspace of P2?c) The set of all vectors of the form a2t2 + a1t1 + a0 (where a2, a1, and a0 are scalar-valued dummy variables) for which a2 + a1 + a0 = 2. Is this set a subspace of P2?In summary, the question is asking whether each of the given sets
  • #1
kaitamasaki
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0

Homework Statement


Which one of the following subsets of [tex]P_{2}[/tex] (degree of 2 or below) are subspaces?
a) [tex]a_{2}t^{2} + a_{1}t + a_{0}[/tex], where [tex] a_{1} = 0[/tex] and [tex]a_{0} = 0[/tex]
b) [tex]a_{2}t^{2} + a_{1}t + a_{0}[/tex], where [tex]a_{1} = 2a_{0}[/tex]
c) [tex]a_{2}t^{2} + a_{1}t + a_{0}[/tex], where [tex]a_{2} + a_{1} + a_{0} = 2[/tex]

Homework Equations


The Attempt at a Solution



First of all I don't even know if the question means only ONE of the three choices is a subspace, or whether I have to decide whether each of them are subspaces or not.

I know that in order for a subset to be a subspace, it must be closed under addition and multiplication.

a) would only have [tex]a_{2}t^{2}[/tex], but doesn't have the other degrees... is it not a subspace then? It is closed under addition ([tex](a_{2}+a_{2})t^{2})[/tex] and and multiplication. Is the question implying the coefficient a2 doesn't change? So you can't have (a2)t^2 + (-a2)t^2 = 0 right?

b) I have all the terms for each degree, so I'm guessing it is a subspace?

c) I have no clue how to do this one. My guess is that I can factor out the 2 and it becomes 2([tex]t^{2} + t + 1[/tex]), but if I add two of these together, I would get 4([tex]t^{2} + t + 1[/tex]), so don't know if that is still in the subspace... does the scalar in front matter or not?
 
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  • #2
Don't guess: appeal to definitions and theorems! They give you a specific list of things to check, right? You say "it must be closed under addition and multiplication". What does that actually mean?


Is that really the exact statement of the problem? It looks very sloppy to me. :frown: What they mean is surely something like:

a) The set of all vectors of the form a2t2 + a1t1 + a0 (where a2, a1, and a0 are scalar-valued dummy variables) for which a1 = 0 and a0 = 0.
 

1. What is a polynomial subset of a subspace?

A polynomial subset of a subspace is a collection of polynomials that satisfy the conditions of a subspace. This means that the polynomials must be closed under addition and scalar multiplication, and must contain the zero polynomial.

2. How are polynomial subsets of subspaces related to linear algebra?

Polynomial subsets of subspaces are an important concept in linear algebra, as they help us understand the structure and properties of vector spaces. They also allow us to apply linear algebra techniques to polynomial functions, which are commonly used in fields such as physics and engineering.

3. How can we determine if a set of polynomials is a polynomial subset of a subspace?

To determine if a set of polynomials is a polynomial subset of a subspace, we must check if the polynomials satisfy the three conditions of a subspace: closure under addition, closure under scalar multiplication, and containing the zero polynomial. If all three conditions are met, then the set of polynomials is a polynomial subset of the given subspace.

4. What are some applications of polynomial subsets of subspaces?

Polynomial subsets of subspaces have many applications in fields such as computer graphics, cryptography, and signal processing. They are also essential in solving differential equations, which have wide-ranging applications in physics, engineering, and economics.

5. Can a polynomial subset of a subspace have an infinite number of polynomials?

Yes, a polynomial subset of a subspace can have an infinite number of polynomials. This is because a subspace can contain an infinite number of vectors, and each vector can correspond to a different polynomial. As long as the polynomials satisfy the three conditions of a subspace, they can form a polynomial subset of the subspace.

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