- #1

fatpotato

- TL;DR Summary
- Proving linear independence of two simple functions in a vector space. Exercice is from a textbook, but solution seems incoherent.

Hello,

I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this :

Prove that each set ##\{f,g\}## is linearly independant in the vector space of all functions ## \mathbb{R}^+ \rightarrow \mathbb{R}##. In this case at point (a) of exercise with ##f(x) = x , g(x) = \frac{1}{x}##

If I understand correctly, I need to show that the only solution to the linear combination of ##f## and ##g## such that ## c_1 f + c_2 g = 0## is the trivial solution, where ## c_1 = c_2 = 0##.

By choosing ##c_1 = 1## and ##c_2 = -1##, I get :

$$ c_1 f + c_2 g = 0 \iff c_1f = -c_2g \iff f = g \iff x = \frac{1}{x}$$

Now, there is indeed a value of ##x## such that this equation is satisfied for non-zero coefficients, with ##x=1##, thus implying that, for at least one value of ##x##, there is a solution to the previous linear combination, so the functions are linearly dependant.

However, the author clearly says that these functions are linearly independant (see correction at http://joshua.smcvt.edu/linearalgebra/jhanswer.pdf#ans.Two.II.1.24). What am I doing wrong? Should I only take the two functions in their most general sense, without evaluating them for a given ##x##?

Thank you.

Edit : spelling

I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this :

Prove that each set ##\{f,g\}## is linearly independant in the vector space of all functions ## \mathbb{R}^+ \rightarrow \mathbb{R}##. In this case at point (a) of exercise with ##f(x) = x , g(x) = \frac{1}{x}##

If I understand correctly, I need to show that the only solution to the linear combination of ##f## and ##g## such that ## c_1 f + c_2 g = 0## is the trivial solution, where ## c_1 = c_2 = 0##.

By choosing ##c_1 = 1## and ##c_2 = -1##, I get :

$$ c_1 f + c_2 g = 0 \iff c_1f = -c_2g \iff f = g \iff x = \frac{1}{x}$$

Now, there is indeed a value of ##x## such that this equation is satisfied for non-zero coefficients, with ##x=1##, thus implying that, for at least one value of ##x##, there is a solution to the previous linear combination, so the functions are linearly dependant.

However, the author clearly says that these functions are linearly independant (see correction at http://joshua.smcvt.edu/linearalgebra/jhanswer.pdf#ans.Two.II.1.24). What am I doing wrong? Should I only take the two functions in their most general sense, without evaluating them for a given ##x##?

Thank you.

Edit : spelling

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