Trivial Solution of a system of equations

• Feb 11th 2013, 09:28 AM
koustav321
Trivial Solution of a system of equations
What do we mean when we say that a system of equations has only a trivial solution i.e x=0,y=0 .... etc. We conclude that these vectors are linearly independent. But what does that visually mean ?

Is it a set of diverging lines from the origin ?

• Feb 11th 2013, 06:19 PM
chiro
Re: Trivial Solution of a system of equations
Hey koustave321.

A linearly independent set of vectors means that each vector can't be represented as a linear combination of the others.

So if I had four vectors a,b,c,d it means that a for example can't be written as a linear combination of b,c,d and so on for all the other vectors.
• Feb 11th 2013, 08:22 PM
koustav321
Re: Trivial Solution of a system of equations
Quote:

Originally Posted by chiro
Hey koustave321.

A linearly independent set of vectors means that each vector can't be represented as a linear combination of the others.

So if I had four vectors a,b,c,d it means that a for example can't be written as a linear combination of b,c,d and so on for all the other vectors.

Thank you for your answer. But may be I am not being able to state my question properly. What i asked,if put in another way, will be -

What do linearly independent vectors 'VISUALLY' mean (not mathematically) ??
• Feb 11th 2013, 10:10 PM
chiro
Re: Trivial Solution of a system of equations
Think about a plane: a set of linearly independent vectors span a plane.

If they were linearly independent then it means that the vector that was linearly dependent would be on the surface of the plane that was formed by the other linearly independent vectors.