Question

How do you check if a set of functions is linearly independent ?

Answer 1

Saw this from an exam last year in a differential equations class. Any ideas on where to begin on this?

Answer 2

usually you want to make a matrix somehow and show that the determinant of that matrix is non-zero

Answer 3

Yeah after googling about linear independence just there, it something do with "wronskian"....I haven't done matrices in a long time haha!

Answer 4

I'm looking at this:
http://tutorial.math.lamar.edu/Classes/LinAlg/LinearIndependence.aspx
example 4, though I don't know what to do about the \(x\in(a,b)=(-1,1)\) part, or what that even has to do with determining linear independence here

Answer 5

it's been a while for me as well :P

Answer 6

Yeah I am watching videos about this wronskian, it's quite confusing and still don't get the point of what are they doing hahaha

Answer 7

check out the link
\[\vec p_1=1\]\[\vec p_2=x\]\[\vec p_3=x^2\]the vector equation is then\[c_1+c_2x+c_3x^2=0+0x+0x^2\]you can do the determinant thingy here, but it's pretty clear by inspection that the only solution is the trivial one.
hence the vectors are linearly independent

Answer 8

I found the solution to my question online, I didn't know the lecturer uploads solutions online, it's just confusing. I'll need to understand everything first before attempting it!!

Answer 9

you could also write\[[\vec p_1|\vec p_2|\vec p_3]=\left[\begin{matrix}c_1&0&0\\0&c_2&0\\0&0&c_3\end{matrix}\right]=\vec0\]and show that\[\left|\begin{matrix}c_1&0&0\\0&c_2&0\\0&0&c_3\end{matrix}\right|=c_1c_2c_3=0\iff c_1=c_2=c_3=0\]hence the trivial solution is the only one, which implies linear independence

Answer 10

not sure if that completely answer the part about the (-1,1) part though

Answer 11

I'll show you the solution to see if you can make sense of it!

Answer 12

ok, so the wronskian for a set of functions \(f_1,f_2,f_3\) is defined as\[W=\left|\begin{matrix}f_1&f_2&f_3\\f_1'&f_2'&f_3'\\f_1''&f_2''&f_3''\end{matrix}\right|\]and if the wronskian is non-zero, that means the functions are linearly independent
that is, they form a fundamental set of solutions to some differential equation

Answer 13

http://tutorial.math.lamar.edu/Classes/DE/FundamentalSetsofSolutions.aspx

http://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx

Answer 14

since \(f_1=1,f-2=x,f_3=x^2\) your solution should now make a bit more sense (hopefully)

Answer 15

\(f_2=x\) *

Answer 16

I'm trying to figure how do I know the matrix is nonzero or not.

Answer 17

do you remember how to take the determinant of a matrix?

Answer 18

http://www.mathsisfun.com/algebra/matrix-determinant.html

Answer 19

a matrix is only zero of each element is zero
we don't care about that here, we only care if the determinant of the matrix is zero

Answer 20

if each*

Answer 21

@turingtest So if I get the determinant of each term in the first row, it should equal zero?

Answer 22

the determinant of the entire matrix should NOT be zero if they are linearly independent