Independant proof 2009c86 1A,B

[nhrock3]

this is a two part question:

v1..vn is a basis

of R^n

and there is a vector b which belongs to R^n and b differs the sero vector

A.

proof that {v1-vn,v2-vn,...,vn-1 - vn}



by definition

in order to prove that a group is independent

we need to show the the only way for



a1(v1-vn)+a2(v2-vn) ..+an(vn-1 -vn)=0



is a1=..=an=0 all the coefficient hs to be zero

so it rang a bell "i need to get a trivial solution "



but trivilal solution is in Ax=0 system could be should if |A| differs zero.



but A is a square matrices by difinition.

how to construct from this single equation a square matrices?

??



the second question:

if v1..vn are solutions to Ax=b system then

rho(A)=1

??

rho(A) is the dimension of row or column space



if v1..vn are solving this system

then the dimension of the solution space is n dim(P(A))=n

and from the formula where n=dim(P(A))+rho(A) we get n=n+rho(A)

so i got

rho(A)=0



but i am asked to prove that rho(A)=1



where is my mistake

 

[lanedance]

this is pretty hard to follow prove what?