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Mathematics 62 Online
OpenStudy (latifah):

in linear algebra: let Ploy2 denote the vector space of polynomials(with real coefficients) of degree less than 3. show that L1={t^2+1,t-2,t+3} and L2={2t^2+t,t^2+3,t}are bases for poly2

OpenStudy (jamesj):

Well, it's clear that {1, t, t^2} is a basis. So any non-degenerate transformation of that basis is also a basis. So if you look at the matrix of coefficients of the vectors in each of L1 and L2, provided that matrix has non-zero determinant, it will follow that they are non-degenerate and therefore also bases.

OpenStudy (jamesj):

the matrix of coefficients of the vectors in each of L1 and L2 with respect to our 'canonical' basis, {1,t,t^2}

OpenStudy (jamesj):

So for example, the first member of L1 has row vector (1,0,1)

OpenStudy (latifah):

ok what about if I want to check span and Linear independent

OpenStudy (jamesj):

that's what I'm saying. The matrix of coefficients has determinant not zero if and if the three vectors are lin. indep which will mean they have 3-d span which means they must span all of Poly2. But if this is using a number of ideas you haven't seen yet, then just do this: Call the member of L1: v1, v2, v3. Show that for any p in Poly2, that there are coefficients c1, c2, c3 such that p = c1.v1 + c2.v2 + c3.v3 I.e., solve the system of equations that this relation gives you and show that it is possible to find such c1, c2, c3.

OpenStudy (latifah):

thank you so much

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