Test the set of functions for linear independence in F

[Sun God]

Homework Statement

Test the set of {1, ln(2x), ln(x^2)} for linear independence in F, the set of all functions.

If it is linearly dependent, express one of the functions as a linear combination of the others.

Homework Equations

N/A

The Attempt at a Solution

I know if [ a(1) + b(ln(2x)) + c(ln(x^2)) = 0 ] implies [ a = b = c = 0 ], then the set is linearly independent. Otherwise, it is linearly dependent.

The book gives an example problem where you can plug in 0 for x and solve for the coefficients, which are then shown to be 0. But that doesn't really work here.

Plugged in x=1: a+b(ln2) + cln(1) =0 implies a + bln2 = 0.

It works neatly in the book example (sinx, cosx) but doesn't really in this case. I can't think of any way to go about doing this...

 

[LCKurtz]

Homework Statement

Test the set of {1, ln(2x), ln(x^2)} for linear independence in F, the set of all functions.

If it is linearly dependent, express one of the functions as a linear combination of the others.

Homework Equations

N/A

The Attempt at a Solution

I know if [ a(1) + b(ln(2x)) + c(ln(x^2)) = 0 ] implies [ a = b = c = 0 ], then the set is linearly independent. Otherwise, it is linearly dependent.

The book gives an example problem where you can plug in 0 for x and solve for the coefficients, which are then shown to be 0. But that doesn't really work here.

Plugged in x=1: a+b(ln2) + cln(1) =0 implies a + bln2 = 0.

It works neatly in the book example (sinx, cosx) but doesn't really in this case. I can't think of any way to go about doing this...

Try simplifying your ln functions using properties of logarithms and see if that gives you any ideas.

 

[Sun God]

Thank you for the help. Apologies again if I'm missing something that's obvious, but I still cannot find the answer. This is my process:

a(1) + b(ln2x) + c(ln(x^2))

= a(1) + b(ln2 + lnx) + c(2lnx)

= a + bln2 + blnx + 2clnx

= a + b*ln2 + (b+2c)*lnx

After trying to play around with it some more, set up some kind of system of equations, reason it out, etc. I still can't find some way to prove or disprove that a = b = c = 0. Any more suggestions? :\

 

[Dick]

Thank you for the help. Apologies again if I'm missing something that's obvious, but I still cannot find the answer. This is my process:

a(1) + b(ln2x) + c(ln(x^2))

= a(1) + b(ln2 + lnx) + c(2lnx)

= a + bln2 + blnx + 2clnx

= a + b*ln2 + (b+2c)*lnx

After trying to play around with it some more, set up some kind of system of equations, reason it out, etc. I still can't find some way to prove or disprove that a = b = c = 0. Any more suggestions? :\

That's good. So you've got a+b*ln(2)=0 and b+2c=0. How about putting c=1 and b=(-2). Now what's a?

 

[Sun God]

Okay, to follow the above poster:

I have a system of equations of

a + bln2 = 0
b + 2c = 0

I set c = 1 and b = -2

Then I get a = 2ln2

Plug this back into the original equation to get

2ln2 - 2ln2 - 2lnx + 2lnx = 0

Which is always true, therefore the functions ARE linearly independent.


My understanding of some of these concepts is probably shaky, since reading through this I still haven't managed to convince myself that this is true (like for instance, how did the values for b and c come about? guess and check?). But I will go over the concepts several times and see if it sticks, thanks everyone!

 

[Dick]

Okay, to follow the above poster:

I have a system of equations of

a + bln2 = 0
b + 2c = 0

I set c = 1 and b = -2

Then I get a = 2ln2

Plug this back into the original equation to get

2ln2 - 2ln2 - 2lnx + 2lnx = 0

Which is always true, therefore the functions ARE linearly independent.


My understanding of some of these concepts is probably shaky, since reading through this I still haven't managed to convince myself that this is true (like for instance, how did the values for b and c come about? guess and check?). But I will go over the concepts several times and see if it sticks, thanks everyone!

No, no. Finding a nonzero a, b, c shows that they are linearly dependent. Not independent!

 

[Sun God]

Ahaha... no wonder it wasn't making sense to me. It's clearer now, much thanks for your patience!

(To clarify for myself and maybe others: it was a matter of finding nonzero coefficients a, b, c and still have the linear combination of the functions equal to zero; if we found such a, b, and c, then the set of functions functions cannot be linearly independent.)

So to finish up the problem, I would show linear dependence by writing one function in terms of the other:

First plug in our values for a, b, c: (2ln2)(1) + (-2)(ln2x) + (ln(x^2)) = 0

which equals:

2ln2 + ln(x^2) = 2ln2x

which we can confirm by playing with ln properties:

2ln2 + 2lnx = 2ln2x
2(ln2 + lnx) = 2ln2x
2ln2x = 2ln2x

Okay. Thanks again.