- #1

Ryker

- 1,086

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**Show this set of functions is linearly independent (e^(-x), x, and e^(2x))**

## Homework Statement

[tex]f_{1}(x) = e^{-x}, f_{2}(x) = x, f_{3}(x) = e^{2x}[/tex]

## Homework Equations

Theorems and lemmas, which state that if vectors are in echelon form, they are linearly independent, and also that they are such if we can find a corresponding matrix, written in echelon form, where the number of rows is the same as the number of original vectors.

## The Attempt at a Solution

I don't really know how exactly to approach this. I guess all of the above functions can be written down in the form akin to a polynomial, such that

[tex]f_{i}(x) = a_{1}e^{-x} + a_{2}x + a_{3}e^{2x}[/tex]

I then put that in matrix form and I basically got

[tex] \left(\begin{array}{ccc}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \\

\end{array}\right) [/tex],

which is a matrix in echelon form that would confim linear independence. Am I even on the right track here? Or are [tex]e^{-x}[/tex] and [tex]e^{2x}[/tex] covered by the same basis vector?

Thanks in advance.

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