Proving That Any Vector in a Vector Space V Can Be Written as a Linear Combination of a Basis Set

[kregg87]

Homework Statement

Show that any vector in a vector space V can be written as a linear combination of a basis set for that same space V.

Homework Equations

http://linear.ups.edu/html/section-VS.html
We are suppose to use the 10 rules in the above link, plus the fact that if you have a lineraly independent set
{X1,X2,...,Xn} then -> c1X1+c2X2+...+cnXn = 0 vector implies that all the constants (c1,c2, etc) are zero.

Not looking for a complete solution, just not sure where to start. I've tried proof by contradiction and a couple other ways and non have worked out for me.

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

Show that any vector in a vector space V can be written as a linear combination of a basis set for that same space V.

Homework Equations

http://linear.ups.edu/html/section-VS.html
We are suppose to use the 10 rules in the above link, plus the fact that if you have a lineraly independent set
{X1,X2,...,Xn} then -> c1X1+c2X2+...+cnXn = 0 vector implies that all the constants (c1,c2, etc) are zero.

Not looking for a complete solution, just not sure where to start. I've tried proof by contradiction and a couple other ways and non have worked out for me.

The Attempt at a Solution

Define "basis". (I know the usual definition, but what is the one YOU are using?)

 

[kregg87]

Define "basis". (I know the usual definition, but what is the one YOU are using?)

My definition is a linearly independent set of N vectors, where N in the dimension of the space. My definition of dimension, N, is it the max number of mutually lineraly independant vectors possible.

 

[fresh_42]

My definition is a linearly independent set of N vectors, where N in the dimension of the space. My definition of dimension, N, is it the max number of mutually lineraly independant vectors possible.

So what can you say about a given vector? Can it be linearly independent from your basis? Or otherwise, what does it mean it can't?