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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 ?

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  • $\begingroup$ 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
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    $\begingroup$ 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
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    $\begingroup$ @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
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    $\begingroup$ @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
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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.

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