6. Let P= {ao+at+at | ao. a,42 R}. That is, P is the linear space...

 

Transcribed Image Text:

6. Let P₂= {ao+at+a₂t² | ao. a₁,42 € R}. That is, P₂ is the linear space of all polynomials of degree less than or equal to two with standard basis 2 = {1, t, t²}. Let W = {f(t) = P₂ | f'(0)=0}. You may assume that this is a subspace of P₂. (a) Show that g(t) = t² W, and h(t) = t & W. (b) Show that the set B = {1, 2} spans W by proving that if a polynomial f(t) = ao+at+a₂t² is in W then a₁ = 0. (c) The set B = {1, 2} spans W. Show that B = {1, 2} is a basis for W. (d) What is dim(W)? You do not need to show work. 6. Let P₂= {ao+at+a₂t² | ao. a₁,42 € R}. That is, P₂ is the linear space of all polynomials of degree less than or equal to two with standard basis 2 = {1, t, t²}. Let W = {f(t) = P₂ | f'(0)=0}. You may assume that this is a subspace of P₂. (a) Show that g(t) = t² W, and h(t) = t & W. (b) Show that the set B = {1, 2} spans W by proving that if a polynomial f(t) = ao+at+a₂t² is in W then a₁ = 0. (c) The set B = {1, 2} spans W. Show that B = {1, 2} is a basis for W. (d) What is dim(W)? You do not need to show work.

Answer rating: 100% (QA)

This question involves linear algebra concepts like subspaces spanning sets and bases within the context of polynomials Lets break down each part of t... View the full answer