Question

Given A is a mxn matrix with dim(N(A)) if u=(α(1), α(2),..., α(n))^T ∈N(A). Prove that α(1)a(1)+α(2)a(2)+...+α(n)a(n)=0,...

Given A is a mxn matrix with dim(N(A)) if u=(α(1), α(2),..., α(n))^T ∈N(A). Prove that α(1)a(1)+α(2)a(2)+...+α(n)a(n)=0, where a(1), a(2),..., a(n) are columns of A.

Now suppose that B is the matrix obtained from A by performing row operations: Show that α(1)b(1)+α(2)b(2)+...+α(n)b(n)=0, where b(1), b(2),..., b(n) are columns of B.

Show that the converse is also true.
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