If I have functions f(x) = sin(x), q(x) = x^2+1 and s(x) = x^2+sin(x) which are linear combinations, how do I represent these functions as vectors? The sin(x) part is throwing me off when constructing the augmented matrix.

yongenelowk

yongenelowk

Open question

2022-08-18

If I have functions f ( x ) = sin ( x ) , q ( x ) = x 2 + 1  and  s ( x ) = x 2 + sin ( x ) which are linear combinations, how do I represent these functions as vectors? The sin(x) part is throwing me off when constructing the augmented matrix.
These functions are linear combinations of a vector space V of continuous functions. I want to find a basis of V. I don't understand how to construct the matrix in the appropriate way to find the basis vector.

Answer & Explanation

Isabella Rocha

Isabella Rocha

Beginner2022-08-19Added 10 answers

These functions each live in the vector space of functions of the form p ( x ) + c sin x for polynomial-or-vanishing p and constant c. For f we take p = 0 , c = 1; for q we take p = x 2 + 1 , c = 0; for s we take p = x 2 , c = 1. In fact, f , q , s live in a smaller space spanned by 1 , x 2 , sin x. You can write them relative to this basis, or any other basis, including of course f , q , s themselves. So, choose whatever basis suits you.

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