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 , wherecis a column vector THIS I GET

(4)then the books says therefore the equation has unique soln forcie. 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