Dgdrxrt

While reading a section on orthogonal matrices, where $vecu_1, ..., vecu_n$ is an orthonormal set of vectors, I find the equation

$$(vecu_1vecu_1^t + ... + vecu_nvecu_n^t)vecx= beginbmatrix
| & ... & | \
vecu_1 & ... & vecu_n \
| & ... & |
endbmatrixbeginbmatrix
-vecu_1^t-\
...\
-vecu_n^t-endbmatrixvecx$$

Now I'm familiar with summarizing a linear combination of vectors by a matrix-vector product, but in doing so I use my concept of the dot product between a row vector in the matrix and the column vector written to the right. This is related to how, in the matrix product, you're basically doing a bunch of dot-products between rows in the left matrix and columns in the right.

Here is something different though, and I'm not sure how I ought to think about it. The columns are written in the left matrix and rows in the right. I don't see how this is consistent with the dot-product view of matrix products. I could chase indices by thinking of the coordinate-wise definition of matrix products, but I feel like the author intends for a better more natural way of understanding this equation.

So is there a good column-row understanding of matrix products, which makes the above equation sensible? Or is the only good way of understanding this by thinking in terms of the coordinate-wise definition of matrix products?

This can be written as
$$
pmatrix vec u_1 & cdots & vec u_n
left[
pmatrix vec u_1^T \ vdots \ vec u_n^T
vec x
right],
$$
where the product in brackets are dot products of $x$ and vectors $u_i$. The resulting numbers are then used to get a linear combination of the vectors $u_i$.