Subsets U of the vector space V

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SUMMARY

The discussion focuses on determining the basis and dimension of the subset U of the vector space V, specifically where V = P3 and U = {p in P3 : p'(0) = p(1)}. The user has established that U is a subspace and is closed under addition and scalar multiplication. By analyzing the polynomial p(x) = a + bx + cx^2 + dx^3, the relationship p'(0) = p(1) leads to the equation a + c + d = 0, which provides a constraint on the coefficients of polynomials in U.

PREREQUISITES
  • Understanding of polynomial spaces, specifically P3.
  • Knowledge of differentiation and its application to polynomials.
  • Familiarity with concepts of subspaces in linear algebra.
  • Ability to manipulate and solve linear equations involving polynomial coefficients.
NEXT STEPS
  • Study the properties of polynomial spaces, particularly P3 and its subspaces.
  • Learn about the basis and dimension of vector spaces, focusing on linear independence.
  • Explore the implications of the Rank-Nullity Theorem in relation to polynomial subspaces.
  • Investigate examples of other polynomial constraints and their corresponding subspaces.
USEFUL FOR

Students and educators in linear algebra, particularly those focusing on vector spaces and polynomial functions, as well as anyone interested in understanding the structure of subspaces within P3.

AkilMAI
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Homework Statement




How can I find the base and dim of U here?, V = P3; U = {p in P3 : p'(0) = p(1)}...

Homework Equations





The Attempt at a Solution


now I've proven it is a subspace and that it is closed under addition and scalar multiplication...but how can I find the base and dim?I was thinking about writing it as p(x)=a+bx+cx^2+dx^4=>p(x)'=b+2cx+3dx^2
and...p'(0)=p(1)=>b=a+b+c+d=>d=-a-c...
Thank you
 
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AkilMAI said:

Homework Statement




How can I find the base and dim of U here?, V = P3; U = {p in P3 : p'(0) = p(1)}...

Homework Equations





The Attempt at a Solution


now I've proven it is a subspace and that it is closed under addition and scalar multiplication...but how can I find the base and dim?I was thinking about writing it as p(x)=a+bx+cx^2+dx^4=>p(x)'=b+2cx+3dx^2
and...p'(0)=p(1)=>b=a+b+c+d=>d=-a-c...
Thank you

Let p(x) be a function in P3, which means that p(x) = a + bx + cx2 + dx3.

For p(x) to be in U, it must be true that p'(0) = p(1), which implies that, and as you found, b = a + b + c + d ==> a + c + d = 0.

Solve for a to get
a = -c - d
b = b
c = c
d = d

The last three equations are trivially true.
From this set of four equations, what can you say about the coefficients of a function p(x) that belongs to U?
 

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