The book starts out writing vectors as row vectors (1 x n matrices), then later introduces the matrix of a vector as a column vector.
Just represent vectors as column vectors from the start, define R^n as R^n,1 (in the notation of the book) and then there's no confusion and you get matrix multiplication of column vectors as students learned in high school: M.x where M is an m x n matrix and x is an n x 1 column vector, with product an m x 1 column vector. Stick to this all the way through a first linear algebra textbook and undergrad tears can be avoided.
e.g. example 3.32 in the text: T(1,0) = (1,2,7). Just write these as column vectors and the matrix multiplication is easier to visualize. (Also, T acting on a real number was written T(x) with brackets so T acting on a vector should really be written T((1,0)). Inconsistencies in notation like this lead to pain.)
Then, for example, when you want to show that the rows of a matrix are independent, you can use a row vector to write succinctly x.M = 0 => x = 0 where x is now a row vector. And it's clear we're doing something other than transforming space vectors here as we've used column vectors for vectors representing points in space.
For undergrad textbooks you want to decide on a convention, one that matches what was done in high school preferably, and stick to it.
You do realize it makes no difference whether the vectors are written horizontally or vertically? Part of mathematical maturity is not getting hung up on pointless notation like this.
Clearly the goal of a first course in linear algebra is not mathematical maturity. If you don't set up the notation consistently from the start, once you start introducing change of bases formulae or proving row rank = column rank, etc., some students will get confused. See all the questions from confused students on Maths Stack Exchange about this point.
The way this particular author has handled the issue is perverse: first he writes vectors as row vectors, then he introduces a mapping M(v) which maps the vector v to its representation as a column vector. Also, as I tangentially noted, he misses out the brackets for a function writing T(0,1) instead of T((0,1)), which is just incompetent. The book should be titled "Algebra Done Shite" not "Algebra Done Right" ;)
Just represent vectors as column vectors from the start, define R^n as R^n,1 (in the notation of the book) and then there's no confusion and you get matrix multiplication of column vectors as students learned in high school: M.x where M is an m x n matrix and x is an n x 1 column vector, with product an m x 1 column vector. Stick to this all the way through a first linear algebra textbook and undergrad tears can be avoided. e.g. example 3.32 in the text: T(1,0) = (1,2,7). Just write these as column vectors and the matrix multiplication is easier to visualize. (Also, T acting on a real number was written T(x) with brackets so T acting on a vector should really be written T((1,0)). Inconsistencies in notation like this lead to pain.)
Then, for example, when you want to show that the rows of a matrix are independent, you can use a row vector to write succinctly x.M = 0 => x = 0 where x is now a row vector. And it's clear we're doing something other than transforming space vectors here as we've used column vectors for vectors representing points in space.
For undergrad textbooks you want to decide on a convention, one that matches what was done in high school preferably, and stick to it.