Hi

How do we work out if the vectors in P2 are linearly dependent or independent.

Thanks

Printable View

- July 23rd 2010, 05:27 AMadam_leedsLinearly independence and dependence of polynomials
Hi

How do we work out if the vectors in P2 are linearly dependent or independent.

Thanks - July 23rd 2010, 05:49 AMHallsofIvy
The most direct way is to use the

**definition**of "linear indepdence" and "linear dependence":

If there exist numbers, , , ..., ,**not all 0**, such that , where that "0" is the 0 vector, then the vectors , , ..., are**dependent**, if not, then they are**independent**.

So you start by setting up the linear combination which, here, is where that "0" is the 0**function**, f(x)= 0 for all x.

Now you can "combine like terms" and use the fact that if a polynomial is 0 for all x then all coefficients must be 0 to get three equations to solve for a, b, and c. Since all three equations will be "= 0", the have the obvious solution a= b= c= 0. If that is**only**solution, then the vectors are independent. It that solution is not**unique**, if there exist other, non-zero, solutions, then they are dependent.

(You don't have to actually find a, b, and c. You can use the fact, if you aready know it, that a system of equations has a unique solution if and only if the determinant of the coefficient matrix is non-zero.) - July 23rd 2010, 05:59 AMadam_leeds
- August 4th 2010, 07:00 AMadam_leeds