By absurd, assume that there is such a linear combination with (1) (that is, not all the scalars are 0). We have that is impossible by the hypothesis ( the others are LI and we have (1) ), so , and therefore that is spans ABSURD!
Let a system of vectors be linearly independent but not generating. Show that it is possible to find a vector such that the system is linearly independent.
So let be any vector that cannot be represented as a linear combination . So does not form a basis for some vector space .
To show linear independence, we have to show that with .