MTH 114 Lecture Notes - Lecture 8: Linear Combination, Linear Independence, Linear Map
Document Summary
Let s be a finite set in a vector space v with the property that every x in v has a unique representation as a linear combination of elements of s. show that s is a basis of v. Two things to prove something is a basis: linear independence 2. spans. The set s spans v because every x in v has a representation as a unique linear combination of elements in s. Suppose s = {v1,, vn} and that c1v1 + + cnvn = 0 for some scalars c1,, cn. The case when c1 = = cn = 0 is one possibility (every subspace has to have the zero vector and this is the only way to have the zero vector) This is the unique and the only possible representation of the zero vector as a linear combination of the elements in s (linear independence means only the trivial solution).
