# Independant proof 2009c86 1A,B

• nhrock3
In summary, to prove that the set {v1-vn, v2-vn, ..., vn-1-vn} is independent, we need to show that the only way for a linear combination of these vectors to equal zero is if all the coefficients are zero. This can be done by constructing a square matrix from the equation Ax=0, where A is a square matrix and x is a vector of coefficients. The second question asks to prove that if v1-vn are solutions to the system Ax=b, then the dimension of the row or column space of A is 1. This can be shown by using the formula n=dim(P(A))+rho(A) and solving for rho(A). However, there seems to
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 independant

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 dimention of row or column space

if v1..vn are solving this system

then the dimention 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

this is pretty hard to follow prove what?

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