I recently tried to build an intuition for why the number of linearly independent rows must equal the number of linearly independent columns after watching the Lecture related to it on go classes. I'm not looking for the formal proof here, but I want to check whether the following reasoning is mathematically sound.
Assume that $A$ is a $5 \times 8$ matrix, and suppose its column rank is $2$. Without loss of generality, let the first two columns be linearly independent, and let every remaining column be a linear combination of these two:
$$
Cj=\alpha_j C1+\beta_j C_2,\qquad j=3,\ldots,8.
$$
Now consider the rows of the matrix:
$$
\begin{bmatrix}
1 & 5 & * & * & * & * & * & * \\
2 & 7 & * & * & * & * & * & * \\
a & b & * & * & * & * & * & *
\end{bmatrix}
$$
Since every column after the first two is determined by the first two columns, the entries marked by $*$ are not independent. Once I choose the first two entries of a row, namely $a$ and $b$, every remaining entry is automatically determined.
For example, if
$$
C3=\alpha C_1+\beta C_2,
$$
then the third entry of every row must be
$$
a\alpha+b\beta.
$$
Similarly, every remaining coordinate is determined by $a$ and $b$. Thus every row has the form
$$
(a,\;b,\;a\alpha3+b\beta3,\;a\alpha4+b\beta4,\;\ldots,\;a\alpha8+b\beta8).
$$
This suggests that although each row is a vector in $\mathbb{R}^8$, it is completely determined by only two free parameters, namely $a$ and $b$. Therefore, all rows lie inside a 2-dimensional subspace of $\mathbb{R}^8$.
Since a 2-dimensional subspace can contain at most two linearly independent vectors, it follows that
$$
\operatorname{rank}{\text{row}}(A)\le2=\operatorname{rank}{\text{column}}(A).
$$
Repeating the same argument after interchanging the roles of rows and columns gives the reverse inequality, so the two ranks must be equal.
My question: Is this intuition mathematically valid? If not, where exactly does it break down? I'm specifically interested in whether the "degrees of freedom" argument above is rigorous enough, or if there is a subtle flaw that I'm missing.
