c = A^-1*v_n+1
i read a proof that there do no exist n+1 linearly independent vectors in a n-dimensional space but i dont quite get some bits
(1)Let be linearly independent vectors in n dimensional space
(2)then must be linearly independent
(3)Hence the matrix A whose columns are is nonsingular ie the only solution to is , where c is a column vector THIS I GET
(4)then the books says therefore the equation has unique soln for c ie. CAN SOMEONE PLEASE EXPLAIN HOW THIS IS, as in how does have a soln and how it's unique??
(5)then it just follows that would be linearly dependent and contary to assumption which i get...
so can someone please please please explain how line (4) came about
that means in a n-dimensional space
you can assume that v_1,...v_n are linearly independent
then v_1,...v_n-1 are linearly independent
then you can find a soln for as you said similiary
which means
and that means the original assumption is wrong as v_1,...v_n will not be linearly independent ???
No. The matrix A will have only n-1 columns and c will have n elements so the product Ac is not defined.
Skalkaz used the result, which you are expected to know, that A is invertible if and only if it's columns are linearly independent.c = A^-1*v_n+1