Vector Spaces in linear algebra

[Fractional-N]

read this Theorem and it's proof
--------------------------------------------

--------------------------------------------
well, I don't understand the proof. I ask 3 questions:
1- In the proof where we used the assumption that m<n?
2- Why "It is possible to choose n scalars ... such that ..." the sum will be zero; and, can you show if "m=n" the will not be zero?
3- Is there any way to determine the n scalars?

simply , I didn't understand the proof, can you explain it to me?

 

[salam2000]

1- this is clear. we use from contradiction that if m>n then subspace isn't independent and is dependence.
2-if for example in XY coordination, if a1X+a2Y=0; then a1=a2=0; also you can set another two basic vector in this cordinate.
3- the Gram schimtd method for find basic univector.