Matrix notation of vectors?

Question

My linear algebra book says that a vector is a one-column matrix. However, in the context of what we are studying (linear equations) it would make more sense if a vector was of the form of the augmented matrix:

$$ \left( \begin{matrix} 1 & 0 & 0 & x \\ 0 & 1 & 0 & y \\ 0 & 0 & 1 & z \end{matrix} \right) $$

This makes more sense to me because a vector is normally represented as an ordered $n-tuple$, and in the example i gave above its clear that the matrix represents the ordered triple $(x, y, z)$. Is the above the formally correct representation of a vector but for simplicity a vector is just represented as a column matrix?

Answer 1

One can view column vectors as $n\times 1$ matrices $$ u = \left( \begin{matrix} x \\ y \\ z \end{matrix} \right) $$ and row vectors as $1\times n$ matrices. $$ u^T = \left( \begin{matrix} x & y & z \end{matrix} \right) $$ No problem here.

You could map your vectors one to one to your given structure, it effectively acts as matrix $(E|u)$, for the identity matrix $E$ and your vector $u$, due to the rule of multiplication of block matrices, but I fail to see the benefit for those extra components. It is a correct representation of a vector but not a minimal one. So I would not use this.

In other contexts such an extension might make sense, for example in computer graphics:

What you provided reminds a bit of a translation matrix for vectors using homogenous coordinates: $$ T = \left( \begin{matrix} 1 & 0 & 0 & x \\ 0 & 1 & 0 & y \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix} \right) $$ Homogenous coordinates have a fourth dimension, which is (most of the time) normalized to $1$, but serve to extend the transformation matrices to $4\times 4$ matrices. Their benefit is to allow to model the important transformations, even the above mentioned translations, as matrix operations. A combined transformation can be obtained by matrix multiplication.

Answer 2

A vector is simply an element of a vector space. You are thinking of vectors in $\mathbb{R}^3$ in terms of their coordinates along the canonical basis $\{e_1,e_2,e_3\}$. From a very formal viewpoint, such a vector should be written as a row, since it represents a co-vector. However, what is $(1,2,3)$? Is it a vector in the vector space $\mathbb{R}^3$ or the co-vector that represents a unique vector along the canonical basis?

Your notation is redundant and actually useless, since the "identity matrix" would be written everywhere without any good reason to do that.