Q: provided that is a linearly independent set of vectors and that the set is linearly dependent, prove that is a linear combination of the 's.

A: Since is linearly independent the vector equation,

has only the trival solution, . Conversely, the set is linearly dependent; thus, there exists a vector in that can be written as a linear combination. Since and is known to be linearly independent we can narrow our search to just one vector, the vector . So, we have a new vector equation,

where

Solving for

can be written as linear comination of 's.

I dunno, I feel a chunck is missing. Is that a sufficient proof?

Thanks