# Thread: matrix of inner product on span(s) with respect to a ordered basis

1. ## matrix of inner product on span(s) with respect to a ordered basis

so i have no idea how to approach this second part of the problem. for the first part, i've figured out that the vectors in the set s are indeed linearly independent. but as far as the second part is concerned, i have no idea what the question even is.
any help is appreciated.
thanks !!

Let V be the vector space of infinitely differentiable functions on R.
1. prove that the set S={e^x, x, x^2} is linearly independent
2. Find the matrix of the inner product <f,g> = integrate from -1 to 1 of f(x)g(x)dx on Span(S) with respect to the ordered basis {e^x, x, x^2}.

heres an image

2. Originally Posted by bombardior
so i have no idea how to approach this second part of the problem. for the first part, i've figured out that the vectors in the set s are indeed linearly independent. but as far as the second part is concerned, i have no idea what the question even is.
any help is appreciated.
thanks !!

Let V be the vector space of infinitely differentiable functions on R.
1. prove that the set S={e^x, x, x^2} is linearly independent
2. Find the matrix of the inner product <f,g> = integrate from -1 to 1 of f(x)g(x)dx on Span(S) with respect to the ordered basis {e^x, x, x^2}.

heres an image
"Linearly independent" means that if $a e^x+ bx+ cx^2= 0$ for all x, then a= b= c= 0. That should be easy to prove.

An inner product, <u, v>, can be written as a matrix by representing u and v as "row" and "column" matrices in <u, v>= uAv. Here, the space is 3 dimensional so A must be a 3 by 3 matrix.

For example, the first basis vector is $e^x$ and the inner product of that with itself would be represented as $\begin{bmatrix}1 & 0 & 0\end{bmatrix}\begin{bmatrix}a_{11} & a_{12} & a_{31} \\ a_21 & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}= a_{11}$ and that must be equal $= \int_{-1}^1 (e^x)^2 dx$ $= \int_{-1}^1 e^{2x} dx$.

In general, [tex]a_{ij}[tex] is the inner product of the "ith" basis vector with the "jth" basis vector.

3. so if i understand correctly, does this just mean i've trying to use a matrix to represent the integral instead of using the inner product definition?
how does the ordered basis come into play?