Subspaces of P spaces and C[a,b] spaces

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In summary, the set of polynomials of degree 3 does not form a subspace of P(4) due to the condition that the leading coefficient must be non-zero. The space of polynomials of degree at most 3 is a subspace, but this is not the same as the space of polynomials of degree 3.
  • #1
seang
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Homework Statement


Determine whether the set of polynomials of degree 3 form a subspace of P(4)

Homework Equations



[tex]P(4) = c_3 x^3 + c_2 x^2 + c_1 x + c_0[/tex]

The Attempt at a Solution



[tex]\alpha P(4_1) = \alpha c_3 x^3 + \alpha c_2 x^2 + \alpha c_1 x + \alpha c_0[/tex]

This just scales the coefficients, right? It would still be a polynomial of degree 3 I think...

For addition, wouldn't you just be adding two polynomials? So wouldn't you just obtain another degree 3 polynomial?

Thanks, Sean.
 
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  • #2
Note that [tex] \Re^{2} [/tex] is not a subspace of [tex] \Re^{3} [/tex]
 
  • #3
I don't see how that helps. I suspect I will, but, why is what you said true? Why don't the set of vectors (a,b,0) doesn't span a subspace in R3?
 
  • #4
Look at vector addition and scalar multiplication in [tex] \Re^{2} [/tex] and [tex] \Re^{3} [/tex]. What are the conditions to show that if [tex] W [/tex] is a nonempty subset of [tex] V [/tex] if [tex] V [/tex] is a vector space over [tex] \Re [/tex] then [tex] W [/tex] is a subspace of [tex] V [/tex]?
 
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  • #5
seang said:
For addition, wouldn't you just be adding two polynomials? So wouldn't you just obtain another degree 3 polynomial?

Thanks, Sean.

Imagine two polynomials of the third degree with leading coefficients a and -a. What happens is you add them?
 
  • #6
courtrigrad said:
Look at vector addition and scalar multiplication in [tex] \Re^{2} [/tex] and [tex] \Re^{3} [/tex]. What are the conditions to show [tex] W [/tex] is a subspace of [tex] V [/tex] if [tex] V [/tex] is a vector space over [tex] \Re [/tex]?




Hmm. Do you mean W is nonempty, the sum of two vectors in W must lie in W, and W times a scalar must lie in W?

So take two vectors in R2, say (x1,x2),(y1,y2). Their sum equals (x1+y1,x2+y2), which lies in R2, and (cx1,cx2) also lies in R2.
 
  • #7
A polynomial of degree 3 is a cubic, ax^3+bx^2+cx+d where a must be non-zero. Thus the space fails to be a vector subspace of the polys for lots of reasons. Note that the space of polynomials of degree at most 3 *is* a subspace, but this is strictly different from the space of polys of degree 3.
 
  • #8
thanks : )
 

1. What is a subspace of a P space?

A subspace of a P space is a subset of the original space that satisfies all the axioms of the P space. In other words, it is a smaller space that still possesses the same properties and structure as the original space.

2. How do you determine if a subset is a subspace of a P space?

To determine if a subset is a subspace of a P space, you must check if it satisfies the three axioms of a P space: closure under scalar multiplication, closure under addition, and contains the zero vector.

3. What is the difference between a subspace of a P space and a C[a,b] space?

A subspace of a P space is a subset that satisfies the axioms of a P space, while a C[a,b] space is a specific type of P space that consists of continuous functions on a closed interval [a,b]. In other words, a C[a,b] space is a subspace of a P space, but not all P spaces are C[a,b] spaces.

4. Can a subspace of a P space be an infinite-dimensional space?

Yes, a subspace of a P space can be an infinite-dimensional space. In fact, many commonly used P spaces, such as C[a,b] spaces, are infinite-dimensional.

5. Are all subspaces of a P space also P spaces themselves?

Yes, all subspaces of a P space are also P spaces themselves. This is because a P space is defined by its axioms, and a subspace that satisfies those axioms is also a P space by definition.

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