In Multiple Regression, how to show the dimension of set $C$ of estimable functions? $C$ is defined as the set of ($k\times 1$) vectors $c$ such that $c^{\top}\beta$ is estimable.
Today I learned that estimable is defined such that the condition must hold: the expectation of a linear combination of the $Y$ equals to a linear combination of $\beta.$
I need to show that the dimension of $C$ is equal to the $\operatorname{rank}(X) = k$.
Here, $X$ is the matrix of the regressors.
Regarding this exercise, I have two questions:

*

*What is actually meant with the dimension of a set?

*How can I show that the dimension of $C$ is equal to the $\operatorname{rank}(X) = k$?


What OP has encountered is nothing but the concept of estimation space aka $\mathcal C(\mathbf X)$ (cf. $\rm [I]$).
Consider an orthornormal basis of $\mathcal C(\mathbf X): ~\langle \boldsymbol\alpha_j\rangle_{j=1}^k.$ Observe
\begin{align}\mathbb E\left[\boldsymbol\alpha_j^\top\mathbf y\right] &=\boldsymbol\alpha_j^\top[\boldsymbol\alpha_1~ \boldsymbol\alpha_2 \cdots ~\boldsymbol\alpha_j ~\cdots~\boldsymbol\alpha_k ]\boldsymbol\beta \\ &= [ 0~0\cdots~ 1~\cdots ~0]\boldsymbol\beta\\ &= \beta_j\tag 1\label 1;\end{align}
For a certain $\boldsymbol{c} = (c_1,\ldots,c_k)\in \mathbb R^k,$ from $\eqref 1, ~\mathbb E\left[c_j\left(\alpha_j^\top\mathbf y\right)\right] = c_j\beta_j.$ Therefore
$$\mathbb E\left[\underbrace{\left(\sum_{j=1}^k c_j\alpha_j\right)^\top}_{\equiv\boldsymbol \ell^\top}\mathbf y\right] =\boldsymbol c^\top\boldsymbol\beta;\tag 2\label 2 $$
$\eqref 2$ implies $\boldsymbol \ell\in \operatorname{span}\{\boldsymbol\alpha_1,~ \boldsymbol\alpha_2, \cdots, ~\boldsymbol\alpha_j, ~\cdots,~\boldsymbol\alpha_k\}.$ For any linear function of parameters to be estimable, there must be $\boldsymbol \ell^\top\mathbf y$ such that $\boldsymbol\ell \in \mathcal C(\mathbf X)$ for which $\eqref 2$ holds.
Hence, the dimension of '$C$' is $k$ as it corresponds to the estimation space.

References:
$\rm [I]$ Plane Answers to Complex Questions: The Theory of Linear Models, Ronald Christensen, Springer Science$+$Business Media, $2011,$ sec. $3.1,$ p. $49$.
$\rm [II]$ Analysis of Variance, Henry Scheffé, John Wiley & Sons, $1959,$ sec. $1.6,$ p. $23.$
