Vtyjkyuk

I am reading this text:

I assume consistency is important here right? Like if you represent one set of addition as a 1xn row, you have to represent another set of addition as a row as well right? You can't represent u+v as a 1xn row and then y+z as a column matrix in the same equation right? Because you can't add a row with a column in matrix addition right?

Exactly, the notations are all equivalent, but (at least to some degree) you have to be consistent with them.

To be more precise, technically, tuples $(x_1,dots,x_n)$, row vectors($1times n$-matrices) $beginpmatrixx_1dots x_nendpmatrix$ and column vectors $beginpmatrixx_1\vdots\x_nendpmatrix$ are all different objects living in different spaces respectively($mathbbF^n$, $mathbbF^(1,n)$, $mathbbF^(n,1)$).

The important thing is, that these spaces are all isomorphic and thus, the different objects can be seen as just a change of notation. This has of course different advantages, like e.g. matrix multiplication is not defined for tuples, but through its identification with row/column vectors(matrices), equations like $Ax=b$ become meaningful.

The way you want or have to represent an object of course always depends on the context. E.g. for your question regarding addition, at least on a formal side, $(x_1,dots, x_n)+beginpmatrixx_1'\vdots\x_n'endpmatrix$ is not defined as they represents objects of different types for which (at least naturally) there is no such thing as addition.

It's common to use $(x_1,x_2,x_3)$ as shorthand for $beginbmatrix x_1 \ x_2 \ x_3 endbmatrix$. Notice the change in the type of parentheses to help avoid confusion.

For the types of row/column matrices, I'm not sure exactly what you are asking. If one addition computation is between $1 times n$ matrices, certainly another computation can involve $n times 1$ matrices. We can add whatever types of matrices we want, and a piece of writing may include several types of matrices.

We do need to be consistent as to whether we are representing vectors in a given space $mathbbF^n$ as column or row vectors, at least if we are going to bring any matrix multiplication into consideration.