why should I transpose "coefficient vector" in linear regression? I was reading a basic state-space model that looks like this (here, X is a vector of predictor variables):
My question is - why is the "coefficient vector" transposed in the regression component? (which seems odd to me) Typically you would see the transpose is made on the "data matrix", like here in linear regression:

or this:

Could anybody please explain?

I don't agree with the notation, but don't get lost in the weeds because I think you can figure out the important bits. Statisticians can get a little loose with their matrix algebra, especially that a matrix with one row or one column is treated like a vector. In several places in the display you posted, I think the author would agree $y + y^T = 2y$.
I am used to seeing the expression for the fitted value as $\hat{y} = X^T \beta$, and consequently $y = \hat{y} + \epsilon$.. The $\beta$ as a matrix is usually thought of as a $p \times 1$ matrix, that's what you get from solving $\hat{\beta} = (X^TX)^{-1}X^T Y$ where $X$ is a $n \times p$ matrix of regressors (with or without an intercept term as a column). Here the $y$ is also thought of as a $n \times 1$ matrix.
