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
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.
the matrix of coefficients of the vectors in each of L1 and L2 with respect to our 'canonical' basis, {1,t,t^2}
So for example, the first member of L1 has row vector (1,0,1)
ok what about if I want to check span and Linear independent
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.
thank you so much
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