Linear Algebra basic notation question

My book writes: A vector in $F^n$ may be regarded as a matrix $M_{n\times 1}(F)$. (true / false)

What is $F$ or $F^n$, and how does the notation $M_{m\times n}(F)$ work? The books also likes to use $M_n(\mathbb{R})$. That is referring to the same basic thing?

Thank you :)

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You have a vector space over a field, usually the field is denoted $F$ (F for Field of course).

$F^{n}$ is the Cartesian product $F\times...\times F$ ($n$ times), for example: the elements of $F^{2}=F\times F$ are all the pairs $(a,b)$ where $a,b\in F$ .

$M_{m\times n}(F)$ denotes all the $m\times n$ matrices with all entries in $F$.

Note that, for example, the elements of $M_{2\times1}(F)$ are of the form $\begin{pmatrix}a\\ b \end{pmatrix}$ where $a,b\in F$ so we can regard this like vectors in $F^{2}$ and vice versa

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Also, $m\times n$ matrices are matrices with $m$ rows and $n$ columns (do you see why a vector is a matrix now?), while $M_n(\mathbb{R})$ should indicate the square $n\times n$ matrices with elements in the field $\mathbb{R}$. – Andrea Orta Sep 14 '12 at 11:18
@AndreaOrta - thanks for adding this – Belgi Sep 14 '12 at 11:19
Thank you for your answer! what exactly do does 'vector space over a field', or 'a vector in $F^n$', or 'matrices with all entries in F' mean? Also I'm not quite sure about what fields are / do. Field explanations on the internet always refer to rings and more, which i heard of, but never learned before.. @AndreaOrta – foaly Sep 14 '12 at 15:52
@foaly - your book did not define what a field is ? or what is a vector space ? if not...change your book IMO – Belgi Sep 14 '12 at 16:33
A field is an algebraic structure, richer than that of a ring and of a group. Important number sets are fields ($\mathbb{Q,R,C}$). That refers to the operations defined on them ($+$ and $\cdot$), with their properties. I think that this is all you have to know about fields for the moment. Now, a vector space must be defined over a field, typically $\mathbb{R\text{ or }C}$. That indicates the set from which all the so called scalars, involved in linear combinations of vectors or entries of matrices, are picked: e.g. if $V$ is defined over $\mathbb{R}$, you won't see complex numbers around. – Andrea Orta Sep 14 '12 at 18:03