In Mathematica:

1. A vector is a list. Example: v = {3, 4, -1, 2}. It has "rank" 1: Part works with 1 index and no more. So v[[2]] is good and v[[2, 1]] gives an error. Also Dimensions[v] is a list of length equal to the rank of v, namely, the list {4} of length 1.

2. A matrix is a list of lists all of the same length. Example: m = {{1, 2}, {3, 4}, {5, 6}}. It has "rank" 2: Part works with 1 or 2 indices and no more. So m[[2]] is good, m[[2, 1]] is good, but m[[2, 2, 1]] gives an error. Also Dimensions[m] is a list of length equal to the rank of m, namely, the list {3, 2} of length 2.

3. We can go on to higher ranks, a list of lists of lists, etc. and the pattern continues. But this would take us away from introductory linear algebra. Such objects are called "tensors." A vector is a rank-1 tensor and a matrix is a rank-2 tensor. I mention it only because some messages in Mathematica refer to vectors and matrices as tensors, and it will be good to be able to understand what such messages are trying to tell you.

Consequently, vectors are not distinguished as row or column vectors. In some languages and in some linear algebra courses, a row vector is a matrix with one row, and similarly, a column vector is a matrix with one column. In Mathematica, you could construct such matrices, but usually you can deal with just vectors. For instance, {{1, 2, 3}} is a row vector, and its transpose {{1}, {2}, {3}} is a column vector, according to the convention I proposed. The following tells how to treat a Mathematica vector as a row or column vector in matrix multiplication:

• For instance m.v multiplies a matrix m and vector v, treating v as a column matrix, assuming you pick a matrix whose number of columns equals the length of the vector.

• And v.m multiplies a matrix m and vector v, treating v as a row matrix, assuming you pick a matrix whose number of row equals the length of the vector.

Finally, if v is a rank-1 vector, then to convert it to a matrix vector you can use ArrayReshape:

rowVector = ArrayReshape[v, {1, Length[v]}]
colVector = ArrayReshape[v, {Length[v], 1}]

The following is another way to convert it:

rowVector = {v}
colVector = List /@ v

As for row vectors versus column vectors, that's really just a convenient way to think about vectors so that the definitions of matrix multiplication extend to vectors. It's a representational choice, but it's not all that substantive from a semantic point of view. A vector of three elements means the same thing whether it's represented as a row vector or a column vector. Mathematica just dispenses with all of that. The List construct is probably the most common thing in Mathematica, and it is used for vectors and matrices. To do matrix multiplication as you see in textbooks, you should use Dot.

v1 = {1, 2, 3};
m1 = {{10, 11, 12}, {20, 21, 22}};
m1 . v1
(* outputs {68, 128} *)

or alternatively

Dot[m1, v1]

This is just a representational choice. There is no distinction in Mathematica between row vector and column vector, so there is no reason to provide a visual distinction between them.