Conditions under which affine independence equivalent to linear independence?

Question

Our professor mentioned in class that for $a \in \mathbb{R}^n$, $b \in \mathbb{R}$, $b \neq 0$ and $x_1, ..., x_k \in \{x \in \mathbb{R}^n\ :\ a^Tx = b\}$, $x_1, ..., x_k$ are affinely independent iff they are linearly independent. I don't have good intuition for what affine independence even represents. Would anyone be able to provide insight into why this is true?

Answer 1

The statement is correct.

If the $x_i$ are affinely dependent, they are also linearly dependent (since the zero sum exhibiting the affine dependence also exhibits a linear dependence).

Conversely, let $\sum_i\lambda_ix_i=0$. Then $a^\top\sum_i\lambda_ix_i=\sum_i\lambda_ia^\top x_i=\sum_i\lambda_ib=b\sum_i\lambda_i$. Since $b\ne0$, it follows that $\sum_i\lambda_i=0$. Thus the linear dependence is also an affine dependence.