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I'm studying linear algebra, I'm reviewing Vector Spaces, and i came across the definition of this Real Vector Space $( \mathcal P_n (\mathbb R), +, \cdot\mathbb R)$:

link to picture 1

where the 'addition' and 'multiplication' operations are defined as follows:

link to picture 2

And I just don't know how to read these definitions, i.e. what do they mean

(of both, the real vector space and the operations)

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  • $\begingroup$ You need understand the definition of vector space, for example, link $\endgroup$ – GAVD May 8 '17 at 6:14
  • $\begingroup$ you can combine a0, b0, because the exponent of x is same for both of them. to be more general, they belong to the same basis. $\endgroup$ – superman May 8 '17 at 6:19
  • $\begingroup$ Please use MathJax instead of images. I've just MathJax-ified your text formula, so you can look at that to see how to do it for the images. Also note that $\forall$ is \forall, $x^n$ is x^n and $\delta$ is \delta. $\endgroup$ – celtschk May 8 '17 at 11:27
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The source of your confusion may be this: the symbols $+$ and $\cdot$ [although suppressed] are being used in two different ways.

First of all we have them in the set of polynomials, just the usual multiplications $a_j x^j$ and so on, and usual additions $a_0+a_1 x +a_2 x^2$ and so on.

But then we want to make this into a Vector Space, so we need to define Vector Addition, and Scalar Multiplication by real numbers. Let's call these operations $\dotplus$ and $\cdot$ for a moment. Then the statement is that $$ (a_0+a_1 x+\dots+ a_n x^n)\dotplus(b_0+b_1 x+\dots+ b_n x^n)$$ is defined to be $$((a_0+b_0)+(a_1+b_1) x+\dots+ (a_n +b_n) x^n)$$ and $$c \cdot (a_0+a_1 x+\dots+ a_n x^n)$$ is defined to be $$(c a_0+c a_1 x+\dots+c a_n x^n).$$

You now need to check that if you do this then the nine (?) axioms for a vector space are satisfied.

Finally, once you are comfortable with what the notation means, you can be a bit more casual. No confusion is introduced since, for example we have that $a_0 \dotplus a_1 x$ (note the plus with a dot) is actually the same by definition as $a_0+a_1 x$.

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The main confusing thing in this definition is "$x$" and its "powers" $x^k$, which are not introduced anywhere, and strictly speaking do not stand for anything in particular. So you should consider the vector "$a_0+a_1x+a_2x^2+\cdots+a_nx^n$" merely as an expression, in which the numbers $a_i$ can be varied but everything else (plus signs and the symbols $x^k$) is fixed, just like in the vector $(a_0,a_1,a_2,\ldots,a_n)$ of the vector space $\Bbb R^{n+1}$ the numbers$~a_i$ van be varied, but the parentheses and commas are obligatory.

In order to have a vector space, one does need to define what addition and scalar multiplication, which is what the second image in the question is somewhat clumsily trying to do. What this amounts to is saying that the vectors in $\mathcal P_n(\Bbb R)$ are added and scalar-multiplied exactly as vectors in $\Bbb R^{n+1}$ are, but adapting for the different notation. So the second line of the image is just the counterpart of the definition $$ (a_0,a_1,a_2,\ldots,a_n) + (b_0,b_1,b_2,\ldots,b_n) = (a_0+b_0,a_1+b_1,a_2+b_2,\ldots,a_n+b_n) $$ of addition in $\Bbb R^{n+1}$, but changing the "decoration" from parentheses and commas to symbols $x^k$ and plus signs. This does cause potential confusion, since many of the plus signs are just dressing up of expressions, while others do correspond to actual addition of vectors (being defined) or of scalars.

All this would be quire weird if there were not some uses of elements of $\mathcal P_n(\Bbb R)$ beyond the pure vector space setting. The "polynomial" notation is suggestive, and allows some abbreviations such as writing $x^2-1$ or even $(x+1)(x-1)$ for what really is $-1+0x+1x^2+0x^3+\cdots+0x^n\in\mathcal P_n(\Bbb R)$. Also one will probably want to consider the operation of replacing $x$ throughout the expression by a number$~a$, and then interpreting everything as real arithmetic operations (including the $x^k$ becoming the $k$-th power $a^k$ of$~a$), so that the whole expression is turned into a simple numeric value; this operation is called "evaluating the polynomial at $x=a$". This operation is not part of what one can generally do in vector spaces, but it is something one can do in $\mathcal P_n(\Bbb R)$ in particular.

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Being too long for a comment: Speaking more generally, if $M$ is a set, then the set $\mathbb{R}^M$ of all real-valued functions defined on $M$ is a vector space. The important point of view is that we want to regard each function $f: M\to \mathbb R$ as a vector. (Actually, this might be sort of well known to you, as for a field $\mathbb F$ the vector space $\mathbb F^n$ should be viewed as the $\mathbb F$-valued functions defined on $\{1,\dotsc,n\}$.)

Now how to define the sum of those vectors? It must be an $\mathbb R$-valued function defined on $M$. A function is defined by the value it takes. So naturally we define for $f$ and $g$ in $\mathbb R^M$ their sum $f+g$ to be $$f+g: \begin{cases} M\to\mathbb R,\\ x\mapsto (f+g)(x):=f(x)+g(x). \end{cases}$$ Similarly the scalar multiplication is defined by $$\delta\cdot f: \begin{cases} M\to\mathbb R,\\ x\mapsto (\delta\cdot f)(x):=\delta\cdot f(x). \end{cases}$$

It's almost trivial to proof that these definition make $\mathbb R^M$ an $\mathbb R$-vector space.

In your special case (with $M=\mathbb R$) you have to consider the subset of $\mathbb{R}^{\mathbb{R}}$ of all polynomials of degree $n$ or less (which is obviuosly closed under addition and scalar multiplication).

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The simplest way how to read this definition is: PnR is the set of polynomials of degree at most n; Operation of addition is the standard addition of polynomials; Operation of scalar multiplication is also the standard multiplication of a polynomial with a scalar.

The meaning is just the fact that PnR with these standard operations satisfies the axioms of a vector space.

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