Easy linear algebra question
Let P2(R) denote the vector space of real polynomial functions of degree less than or equal to two and let B := [p0,p1,p2] denote the natural ordered basis for P2(R) (so pi(x) = xi). Define g ∈ P2(R) by g(x) = 2x2 − 3x + 1. Write g as a linear combination of the elements of B. Compute the coordinate vector gB of g with respect to B.
Define h1,h2,h3 ∈ P2(R) by h1(x) = 2, h2(x) = 3x−2 and h3(x) = 2x2 −3x+1. Define C := [h1, h2, h3]. Write each element of B as a linear combination of the elements of C. Explain why the calculations you have performed prove that C is a basis for P2(R).
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Re: Easy linear algebra question
What you have written is very difficult to understand. Does "x2" mean x2? I have no idea what "pi(x)= xi" means. My first guess would be pi(x)= xi so that p0= 1, p1= x, and p2= x2. Is that right?
If so then "the coordinate vector" is just something like (A, B, C) where A is the coefficient of 1, B is the coefficient of x, and C is the coefficient of x2.
"Write each element of B as a linear combination of the elements of C" means to find a, b, c such that 1= a(2)+ b(3x- 2)+ c(2x2- 3x+ 1), d, e, f, such that x= d(2)+ e(3x- 2)+ f(2x2- 3x+ 1), and g, h, i, such that x2= g(2)+ h(3x- 2)+ i(2x2- 3x+ 1).