I am trying to understand the difference between scalars and vectors. I know the basic definition that vectors are magnitude with direction. But we also call vectors those entities that belong to vector spaces and can be added.

I had previously asked this question, a comment to this answer to the question states that real numbers can be both vectors and scalars, how is that even possible ?

If so then what is a vector and what is a scalar ? How can they be different but be represented simultaneously by the same mathematical concept ?

  • $\begingroup$ It is possible if you define it accordingly. So we need your definition for a vector space to tell you why the real numbers can take both the role of a space and the role of scalars in this setting. Your intuitive understanding of vectors with direction etc. only classifies a small set of vector spaces, there are many more out there. $\endgroup$ – Dirk Jun 13 '17 at 8:23
  • $\begingroup$ There is a difference between real scalars, which are elements of $\mathbb R$, denoted as $r$, and $1$-tuples of reals (1D vectors), which are elements of $\mathbb R^1$, and denoted as $[r]$. In 1D, the magnitude is just the absolute value, and only two directions are possible: $[-1]$ and $[+1]$. But both $\mathbb R$ and $\mathbb R^1$ are vector spaces. $\endgroup$ – Yves Daoust Jun 13 '17 at 8:26
  • 4
    $\begingroup$ The difference between a vector and scalar is contextual. To be more precise, if $V$ is a vector space over a field $F$, then the elements of $V$ are called "vectors" and the elements of $F$ are called "scalars". Since $\mathbb{R}$ is a vector space over itself, real numbers are both vectors and scalars in this context. In other contexts, a real number might be a vector but not a scalar (e.g. the vector $\sqrt{2}$ in $\mathbb{R}$ over $\mathbb{Q}$), a scalar but not a vector (e.g. the scalar $1$ in $\mathbb{R}^2$ over $\mathbb{R}$), or neither a vector nor a scalar. $\endgroup$ – diracdeltafunk Jun 13 '17 at 8:27
  • 2
    $\begingroup$ @ng.newbie This is correct. "magnitude with direction" is not a valid definition of "vector", in the mathematical sense. However, it is often taught to those taking intro physics classes or the like. In general, a vector is just an element of a vector space. Have you seen the definition of a vector space before? $\endgroup$ – diracdeltafunk Jun 13 '17 at 8:29
  • 1
    $\begingroup$ @diracdeltafunk No I have not. And yes I am interested in purely the mathematical definition of a vector and not anything else. I want to know hoe pure maths defines and handles them. Would be great if you could point out other mathematical literature I need to understand before I get to making sense of vectors, because right now nothing about them is making sense. $\endgroup$ – ng.newbie Jun 13 '17 at 8:32

A vector space is a quadruple $\{V,\mathbb{F},+ ,\odot\}$ where $V$ is a set whose elements are called vectors, $\mathbb{F}$ is a field whose elements are called scalars, $+:V \times V \to V$ is a binary operation in $V$, called addition, and $\odot : \mathbb{F}\times V \to V$ is an operation called ''scalar multiplication'' (usually the symbol $\odot$ is omotted). These operations must satisfies some axioms that ''define'' what a vector space is.

So If we take $V=\mathbb {R}$ and define the addition as the usual addition in the field of real numbers, ve can build some vector spaces in which the real numbers are vectors.

If we chose $\mathbb{F}=\mathbb{R}$ we have a vector space in which also the scalar are real numbers: the so called vector space $\mathbb{R}$ over $\mathbb{R}$. Ve can prove that this is a vector space of dimension $1$ and we can think, intuitively, to the vectors of this space as the point of a straight line.

If we chose $\mathbb{F}=\mathbb{Q}$ we have a vector space in which the scalar are the rational numbers. This is the vector space $\mathbb{R}$ over $\mathbb{Q}$. and it is very different as a vector space because its dimension is uncountable infinite.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.