Why are the rows of a parity-check matrix linearly independent? I've been reading about error-correcting codes, and I came across the following definition for a parity-check matrix: 
''There is an $(n-k) \times  n$ matrix $H$, called a parity check matrix for an $[n,k]$ linear code $\mathcal{C}$, defined by
$$\mathcal{C} = \{x \in \mathbb{F}^{n}_{q} | Hx^{T} = 0 \}.'' $$
Then, the book states that the rows of $H$ will also be linearly independent, but I am having trouble seeing why this is true. 
As of now, I've noticed that the map $x \mapsto Hx^{T}$ is a linear transformation from $\mathbb{F}^{n}_{q}$ to $\mathbb{F}^{n-k}_{q}$ with $\mathcal{C}$ being the kernel of this linear transformation, but I can't make the connection to linear independence. 
Can anyone help?

I think Dilip has it (in a comment above). Remember the Rank-Nullity theorem for this one. ($\mathcal{C}$ is the kernel of the linear transformation described by $H$)
