Linear dependence of a set of linear combinations If the elements of $A=\{a_1,a_2,...,a_n\}$ are linear combinations of the elements of another set, say $B=\{b_1,b_2,...,b_n\}$, then how can I show that $A$ is not linearly independent?

You can't because that statement is not always true. Counterexample: If $B = \{ (1,0,0), (0,1,0), (0,0,1) \}$, then $A = \{ (1,1,0), (1,0,1), (0,1,1) \}$ is a set of linear combinations of the elements of $B$, but $A$ is linearly independent. 
