Row and column vector representation for Euclidian vector spaces

Question

When dealing with Euclidean vector spaces (such as those in physics) we consider that a vector $A$ with a tuple of scalar components $(a,b,c)$ is equal to a vector $B$ with the same components, however, we make a distinction between column and row vectors (namely that one is the transpose of the other) is it that for a euclidean vector we just decide on one way of 'representation' and the distinction is more for dealing with matrices? Because this suggests that given a row vector $A$ and Column vector $B$ with the same components they are not equal. Is it a form of notation for vectors or are we essentially writing the components in a matrix?

If we define a vector $a=(a_1,a_2)$ as being represented as a column and row vector, then for $a$, $Ma≠Ma$ as one is defined and the other is not.

Answer 1

Let $n$ be a positive integer.

• A row vector with $n$ real components is a matrix with one row and $n$ columns, i.e. an element of $\mathbb R^{1\times n}$.
• A column vector with $n$ real components is a matrix with one column and $n$ rows, i.e. an element of $\mathbb R^{n\times 1}$.
• An element of $\mathbb R^n$ is a list of $n$ real numbers.

Of course we identify lists, row and column vectors through "obvious" bijections. For example, given a list$$(x_1,\ldots,x_n)\in\mathbb R^n$$ we can set $$\forall i:x_{i,1}:=x_i$$and $$\begin{pmatrix}x_{1,1}\\\vdots\\x_{n,1}\end{pmatrix}\in\mathbb R^{n\times 1}$$is the column vector associated to the list.

Answer 2

You are confusing notation and coordinate representation of vectors. That is, a vector space is an algebraic structure defined using four objects $(V, \mathbb{F}, \oplus ,\odot )$ where $(V, \oplus)$ is an Abelian group, $(\mathbb{F},+,\cdot )$ is a field and

$$ \odot :\mathbb{F}\times V\to V,\quad (\lambda ,\mathbf{v})\mapsto \lambda \odot \mathbf{v} $$

is a function named scalar multiplication, such that the following conditions holds:

$$ \lambda \odot (\mathbf{v}\oplus \mathbf{w})=(\lambda \odot \mathbf{v})\oplus (\lambda \odot \mathbf{w}),\quad 1\odot \mathbf{v}=\mathbf{v},\quad 0\odot \mathbf{v}=\mathbf 0 \\(\lambda +\mu)\odot \mathbf{v}=(\lambda \odot \mathbf{v})\oplus (\mu \odot \mathbf{v}),\quad (\lambda \cdot \mu )\odot \mathbf{v}=\lambda \odot (\mu \odot \mathbf{v}) $$

If the vector space have finite dimension (say dimension $n$), then there exists some list of linear independent vectors $\mathbf{v}_1,\ldots ,\mathbf{v}_n$ such that for every $\mathbf{w}\in V$ there exists scalars $\lambda _j\in \mathbb{F}$ such that

$$ \mathbf{w}=(\lambda _1\odot \mathbf{v}_1)\oplus \ldots \oplus (\lambda _n\odot \mathbf{v}_n) $$

Then, using the previous list as a basis of $V$ we can represent the vector $\mathbf{w}$ as $(\lambda _1,\ldots ,\lambda _n)$, that is, there is a bijective map $\phi :V\to \mathbb{F}^n$ such that to each vector in $V$ gives a coordinate representation $(\lambda _1,\ldots ,\lambda _n)$, what is an element of $\mathbb{F}^n$.

So, for any Euclidean space $V$ of dimension $n$ we directly use the coordinate representation given by elements of $\mathbb{R}^n$. Now, as a notation we can represent any element of $\mathbb{R}^n$ by the standard notation for $n$-tuples, that is $(\lambda _1,\ldots ,\lambda _n)$, or using a matrix-like vertical notation

$$\begin{pmatrix} \lambda _1\\ \vdots \\ \lambda _n \end{pmatrix}$$

However both notations represent the same vector $\mathbf{w}\in V$, but we choose some or other notation depending on the context to make things easier, by example if $M$ is an $n\times n$ matrix then we choose the vertical notation to represent the action of $M$ by the left to some vector, in this case

$$ \begin{pmatrix} M_{1,1}&&\cdots &&M_{1,n}\\\vdots && &&\vdots \\M_{n,1}&&\cdots && M_{n,n} \end{pmatrix}\begin{pmatrix} \lambda _1\\ \vdots \\ \lambda _n \end{pmatrix} $$

However, to write the coordinates of $\mathbf{w}$ inside a text is preferable to use the notation $(\lambda _1,\ldots ,\lambda _n)$ instead. I hope you see it more clear now.

Answer 3

Given a vector space, V, over field F (typically R or C) we can define its "dual space", V*, as the set of all linear functions from V to F. It can be shown that V and V* are isomorphic so V* is also a vector space.

In particlar, if V is finite dimensional, say dimension n, it is customary to represent each vector as a vertical n-column and each linear function as a horizontal n-row. For example, if V is three dimensional we can write a vector as $v= \begin{bmatrix}x \\ y \\ z \end{bmatrix}$ and a linear function as $f= \begin{bmatrix}a & b & c \end{bmatrix}$. Then the operation of function f on vector v can be written as a matrix product: $f(v)= \begin{bmatrix}a & b & c \end{bmatrix}\begin{bmatrix}a & b & c \end{bmatrix}= ax+ by+ cz$.

This is, of course, just notation.

Answer 4

The answers above already well clarify that there is an isomorphism between lists (n-uples), row vectors and column vectorsand thus the choice is a matter of convenience.

In linear algebra the column notation is more common because more conforming to the way we write systems of linear equations.
Instead the row notation is common practice in Markov chains just because the matrix then looks tha same as the usual representation of a transition matrix as row $\to$ column.

But interestingly, since you are studying physics, later on you will face with the important concept of co-variant and contra-variant vectors, dealing with which is useful to use both conventions together, while keeping them distinct.