# Definition of linearly independent

1. May 9, 2010

### Zoe-b

I've looked it up and can't really find a clear answer..
For a system of 3 simultaneous linear equations, is there any difference between 'the equations are linearly independent' and 'the equations have a unique solution'. If so what is it?
Thanks!

2. May 9, 2010

### hgfalling

Two or more equations are linearly independent if they are:

a) linear (ie, no terms with power > 1 and no multiplication of variables by each other),
b) independent (none of the equations can be derived algebraically from the others).

Some examples:

Linearly independent:
2x + 3y = 8
3x + 2y = 5

x + y + z = 7
3x + 2y = 17

Not linearly independent:
x + y = 4
2x + 2y = 8
(second equation is a multiple of the first)

x+y=3
x+2y=9
2x+3y=12
(third equation is sum of the first two)

x2 + y = 1
x + 3y = 9

xy + 3y = 6
(xy term makes equation not linear)

A system of linearly independent equations need not be consistent, but if the left-hand sides of all the equations are linearly independent, then it always will be.

3. May 9, 2010

### Zoe-b

Hmmn.. thanks that does help.
I guess what I really meant is if a system of equations has a unique solution then is it always linearly independent?

4. May 9, 2010

5. May 9, 2010

### D H

Staff Emeritus
Condition (a) is not required. For example, the functions f(x)=x and g(x)=x2 are linearly independent.

Condition (b) is not quite correct. Better said, a set of expressions {f1(x1,x2,...), f2(x1,x2,...), ...} are linearly independent if the only solution to a1f1+a2f2+...=0 is the trivial solution a1=a2=...=0.