Linear independence of basis vectors

[cappygal]

How do I prove the linear independence of the standard basis vectors? My book is helpful by giving the definition of linear independence and a couple examples, but never once shows how to prove that they are linearly independent.
I know that the list of standard basis vectors is linearly independent if:
The only choice of a_1, a_2, ... a_m that makes a_1v_1+a_2v_2+...+a_mv_m=0 is a_1=a_2=...=a_m=0.
But i don't know where to go from there .. any help would be appreciated

 

[hypermorphism]

Use contradiction. If they were linearly dependent, then by your definition, at least one of the vectors could be written as a linear combination of others. Is this true of the standard basis ?

 

[TD]

For example, in $\mathbb{R}^3$, the standard basis is:

$$\left\{ {\left( {1,0,0} \right),\left( {0,1,0} \right),\left( {0,0,1} \right)} \right\}$$

This basis is linearly independent if, as you say:

$$a\left( {1,0,0} \right) + b\left( {0,1,0} \right) + c\left( {0,0,1} \right) = \left( {0,0,0} \right)$$

implies that $a = b = c = 0$.

Well, solve the condition for a, b and c and see if you can find anything else besides the solution we expect.
Can you now see how it will be for $\mathbb{R}^n$ in general?

 

[cappygal]

Thank you so much .. it actually makes sense now .. Something about being out of school with a broken pelvis means that it's harder to understand what they do in class without you ... thanks!

 

[TD]

No problem