Definition of linearly independent

[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!

 

[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
(equation is quadratic, not linear)

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.

 

[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?

 

[Mark44]

Yes, and it goes the other way, too. I.e., if the system is linearly independent, then there is a unique solution. Here's a link to some information that might be helpful - http://en.wikipedia.org/wiki/System_of_linear_equations.

 

[D H]

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).

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.