- #1

- 433

- 7

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter MathewsMD
- Start date

- #1

- 433

- 7

- #2

Stephen Tashi

Science Advisor

- 7,642

- 1,495

A common misconception is thatW= 0 everywhere implies linear dependence.

- #3

Stephen Tashi

Science Advisor

- 7,642

- 1,495

Writing the jth column of [itex] M_{*.j} [/itex]

[itex] M\ x = x_1 M_{*,1} + x_2 M_{*.2} + ... x_n M_{*,n} [/itex]

If we have more or fewer unknowns than equations, the matrix [itex] M [/itex] isn't square, so we can't do an analysis by taking its determinant. When [itex] M [/itex] is a square matrix and its derminant is non-zero, we can find a unique solution for variables [itex] x [/itex]. In particular , we can find a unique solution for the system [itex] M\ x = b [/itex] when [itex] b [/itex] is the column vector of zeroes. The unique solution to [itex] M\ x = 0 [/itex] would be [itex] x_j = 0 [/itex] for all [itex] j [/itex]..

If the column vectors of [itex] M [/itex] were dependent then solution for [itex] M\ x = 0 [/itex] would not be unique. For example if [itex] M_{*,1} = \sum_{j=2}^n a_j M_{*,j} [/itex] with at least one of the [itex]a_j [/itex] nonzero then the values [itex] x_1 = 1 [/itex] and [itex] x_j = -a_j [/itex] for [itex] j > 1 [/itex] would be a nonzero solution to [itex] M\ x = 0 [/itex].

By analogy, the Wronskian [itex] W [/itex] is the derminant of a square matrix of functions [itex] M [/itex] . A column vector of [itex] M [/itex] gives a function and it's successive derivatives. If [itex] W [/itex] is nonzero then [itex] M\ x = 0 [/itex] has the unique solution [itex] x= 0 [/itex] , so the column vectors of [itex] M [/itex] are independent.

That's just "by analogy". There would be lots of technicalities to consider if we want to prove anything about solutions to a differential equation. At least the analogy reminds us that a given column has entries all relate to the same function ( and a given row has entries that all relate to the same order of differentiation).

Share: