Linearly independent columns of a matrix product Given $\mathbf{A} = \mathbf{B}\mathbf{C}$, with $\mathbf{B} \in \mathbb{R}^{m\times n}$ and $\mathbf{C} \in \mathbb{R}^{n\times p}$. Say we know that the columns of $\mathbf{A}$ are linearly independent. Does this also imply that the columns of $\mathbf{B}$ and $\mathbf{C}$ are linearly independent? How can I go about proving this?
We know $\text{Ker}(\mathbf{BC}) = \left\{\mathbf{0}\right\}$, so $\mathbf{x} = \mathbf{0}$ is the only vector satisfying 
$$
\mathbf{BCx} = \mathbf{0}
$$
From this:
$$
\mathbf{BC}(\mathbf{0}) = \mathbf{B}\mathbf(0) = \mathbf{0}
$$
but this only shows that $\mathbf{0}$ is in the null space of $\mathbf{B}$, not that it's the only member of the nullspace.

The column vector of $C$ must be linear independent. You might try to show that if $A = BC$ are just mappings and $A$ is injective, then so is $C$. However, $B$ might not be injective.
