Prove range$(A)$ is the space spanned by the columns of $A$

Question

Prove range$(A)$ is the space spanned by the columns of $A$.

Since both of these things are sets we traditionally show that both sets are a subsets of the each other. But why doesn't the following proof work?

proof

Let $A \in \mathbb{C}^{m\times n}$. Then range$(A)$ is the set of all vectors $Ax$, $x \in \mathbb{C}^n$. Furthermore $Ax$ can be written

$$Ax = x_1a_1 + \cdots x_na_n \tag{$\star$}$$

where $x_i$ are the components of the vector $x$ and $a_i$ are the columns of $A$. Since $x$ is arbitrary so are its components. Therefore the right side of $(\star)$ represents all possible linear combinations of the column vectors $a_i$. That is

$$\text{range}(A) = \{Ax: x \in \mathbb{C}^n\} = \text{span}\{a_1, \dots a_n\}$$

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Edit: Note that the book I'm studying out of basically does what I do above but then includes a "conversely" statement that starts by letting some vector be in the span of the columns of $A$. That's why I think mine is wrong.

Answer 1

Your proof is correct. $\ $