Consider the subset U ⊂ R3[x] defined as

  • Thread starter Thread starter Karl Porter
  • Start date Start date
Click For Summary

Homework Help Overview

The discussion revolves around the subset U of R3[x], specifically focusing on proving that it is a subspace of polynomials of degree 3 or lower that have roots at x=0 and x=-1. Participants are exploring the definitions and properties of vector spaces in relation to polynomial functions.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to prove closure under addition and scalar multiplication for the subset of polynomials. Questions arise regarding the implications of substituting values into the polynomials and the nature of their roots. There is also discussion about the distinction between polynomial operations and vector space operations.

Discussion Status

Participants have engaged in a back-and-forth dialogue, clarifying concepts and exploring the properties of polynomials within the context of vector spaces. Some guidance has been offered regarding the nature of polynomial addition and scalar multiplication, but no consensus has been reached on the completeness of the proof.

Contextual Notes

There is an ongoing examination of the definitions and properties of polynomial functions and vector spaces, with participants questioning assumptions about the operations involved. The discussion reflects a mix of understanding and uncertainty regarding the requirements for proving subspace properties.

  • #61
Mark44 said:
The part above is OK.
No. Take a closer look at post #56. I've laid it all out for you, including how to get a set of basis functions.
I have no idea what you're trying to say here.
Basis would be {x^3+x^2,x^3-x}
 
Physics news on Phys.org
  • #62
Karl Porter said:
Basis would be {x^3+x^2,x^3-x}
Yes.
Let's call these ##p_1(x)## and ##p_2(x)##, respectively.
As a check that these functions are in U, it would be good to verify that ##p_1(0) = 0, p_1(-1) = 0## and that ##p_2(0) = 0, p_2(-1) = 0##.
 
  • Like
Likes   Reactions: Karl Porter

Similar threads

Understanding Subspaces: Criteria and Examples (R3, Integers, R2)
  • · Replies 5 ·
Replies
5
Views
2K
Null space of dual map of T = annihilator of range T
  • · Replies 1 ·
Replies
1
Views
2K
Determining if a subset W is a subspace of vector space V
  • · Replies 8 ·
Replies
8
Views
2K
Determine if function forms a vector space
  • · Replies 3 ·
Replies
3
Views
2K
Proving that a subset is a subspace
  • · Replies 4 ·
Replies
4
Views
2K
Define the notation used here in describing a series?
  • · Replies 8 ·
Replies
8
Views
3K
[Linear Algebra] Another question on subspaces
  • · Replies 15 ·
Replies
15
Views
2K
Constructing a Non-Subspace in R^2 with Closed Addition and Additive Inverses
  • · Replies 9 ·
Replies
9
Views
2K
Proving Kernel of T is a Subspace of V
  • · Replies 9 ·
Replies
9
Views
3K
Determining a subspace of polynomials with degree 3
  • · Replies 2 ·
Replies
2
Views
18K