Consider the 2 way ANOVA model:
$y_{i j} = \mu + l_j + \alpha_i + e_{i j}$, $e_{i j} \sim \mathcal{N} (0, \sigma^2)$
where $i=1\dots m, j=1\dots n$, how do you prove that the residual degrees of freedom is $(n - 1)(m - 1)$? I can sort of see this intuitively, as the model can still be fitted exactly when $l_1 = 0, \alpha_1 = 0$, but how would I prove this rigorously (I think it's defined using the dimension of the vector space spanned by the coefficients on RHS)?
To show this rigorously, we should view the $mn$ observations $y_{ij}, i = 1, \ldots m,\; j = 1, \ldots, n$ as an $mn$-vector in the vector space $\mathbb{R}^{mn}$. By the RHS specification of the model, the linear part is spanned by vectors: $$e, r_1, \ldots, r_m, c_1, \ldots, c_n$$ where $e$ is the $mn$-vector of all ones. $r_i$ is the $mn$-vector with its $(i - 1)m + 1, \ldots, (i - 1)m + n$ components $1$ and all else zeros (known as "row vectors"), and $c_j$ is the $mn$-vector with its $j, n + j, \ldots, (m - 1)n + j$ components $1$ and all else zeros (known as "column vectors").
We may denote the linear subspace spanned by these vectors by $\mathcal{L}$. It is easily seen $$\dim(\mathcal{L}) = 1 + (m - 1) + (n - 1) = m + n - 1.$$
Since the residual degrees of freedom is understood as the dimensionality as the orthogonal complement subspace of $\mathcal{L}$ in $\mathbb{R}^{mn}$, we have: $$df_{\text{residual}} = \dim(\mathbb{R}^{mn}) - \dim(\mathcal{L}) = mn - (m + n - 1) = (m - 1)(n - 1).$$
Remark: why $R \cap C = \text{span}(e)$? You may interpret the intersection of two subspaces just in the sense of intersection of two ordinary sets. Rigorously, I should write $R \cap C = \text{span}(e)$ instead of $R \cap C = e$ (sorry for any confusion).
To show this, first, by definition, $e \in R \cap C$. On the other hand, any vector belongs to $R$ has the form $\alpha_1 r_1 + \cdots + \alpha_m r_m$ (this is because, as I argued in the comment, for example, $R$ is in fact spanned by $\{r_1, \ldots, r_m\}$, which is known as a "basis" of $R$) whereas any vector belongs to $C$ has the form $\beta_1 c_1 + \cdots + \beta_n c_n$, so if $x \in R \cap C$, then $x = \alpha_1 r_1 + \cdots + \alpha_m r_m = \beta_1 c_1 + \cdots + \beta_n c_n$. By carefully writing down these two representations and equalizing them, you can deduce that $$\beta_1 = \alpha_1 = \cdots = \alpha_m, \beta_2 = \alpha_1 = \cdots = \alpha_m, \ldots, \beta_m = \alpha_1 = \cdots = \alpha_m$$ Thus all the $\alpha, \beta$ coefficients must be identical, let the common value be $c$, then $$x = c(\alpha_1 + \cdots + \alpha_m) = c(\beta_1 + \cdots + \beta_n) = ce \in \text{span}(e).$$
In summary, we have shown that $$\text{span}(e) \subset R \cap C \subset \text{span}(e).$$
Therefore, $R \cap C = \text{span}(e)$.
