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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 $\mathbb{R}$.2012-09-14
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    @AndreaOrta - thanks for adding this2012-09-14
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    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.. @AndreaOrta2012-09-14
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    @foaly - your book did not define what a field is ? or what is a vector space ? if not...change your book IMO2012-09-14
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    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.2012-09-14
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    By the way, I see you have difficulties about very basic concepts. But I understand you are just at the beginning of a course in linear algebra. So, don't panic, but study hard, paying great attention to the initial definitions, so that you can learn the essential vocabulary and use it fluently. I can't believe that the things we talked about here aren't explained in your book, maybe you just have to go back to the very first pages! Should you have problems in the future, feel free to ask. :)2012-09-14
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    the point is, i don't have my book yet. just copies from the pages with the homework so far. my university is on a different continent than my highschool, and so the profs use things without mentioning them, that i never heard of before (different highschool education). So - yea. as soon as i got the books (wed), i'll go through the first pages for sure :P by the way, our books are: Friedberg, Insel and Spence, Linear algebra, 4th edition && R.A.Adams: Calculus, a Complete Course, 7th ed. @AndreaOrta2012-09-16