If $\displaystyle A$ is a Hamel basis for a vector space $\displaystyle X$ with $\displaystyle n$ elements, then every subset of $\displaystyle X$ having $\displaystyle n+1$ elements are not linearly independent.

Proof:

I am doing this by inductions. I've shown it's true when $\displaystyle n=1$.

Assume it's true when $\displaystyle n = k$.

I have trouble with the case when $\displaystyle n = k+1$.

Let $\displaystyle A = \{x_1, ..., x_{k+1}\}$ be a Hamel basis, and $\displaystyle B=\{y_1, ..., y_{k+2}\}$ be a subset of X. Let $\displaystyle [A]$ be the collection of all linear combinations of $\displaystyle \{x_1, ..., x_k\}$.

How do I show that B is not linearly independent?