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.
As it was said before, A must not be lineary dependent. The easy way to define it is to make a matrix, where coefficients are the coefficients from linear combinations of vectors from A through B, and then calculate its determinant. If det = 0 $ \Rightarrow$ A is not linearly independent. For example, If B={(1,0,0),(0,1,0),(0,0,1)}, A={(1,1,1),(1,0,1),(0,1,0)} then  matrix is \begin{pmatrix} 1 &amp; 1 &amp; 1 \\ 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \end{pmatrix}, it's det = 0, so A is not lineary independent