What is a linear combination, exactly?
Andrew Henderson 
I'm used to the definition of linear combination used in linear algebra textbooks. I'm reading the book Algebra by Artin and on page 357 he says:
If $R$ is the ring $\mathbb{Z}[x]$ of integer polynomials, the notation $(2,x)$ stands for the ideal of linear combinations of $2$ and $x$ with integer polynomial coefficients.
According to the definition above, the term $x\cdot x+2 = x^2+2$ is a linear combination of $x$ and $2$, which doesn't feel like linear to me.
Here is my question: What is the definition of a linear combination?
My confusion may arise because in a vector space there is no such thing as product of vectors.
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$\begingroup$Let's look at a finite set of things, like $\{ x, y, z, w \}$. Let's pretend they are just variables. If we want to talk about a linear combination of these things, we first need to know what our "scalars" will be.
If we take our scalars to be all real numbers $\Bbb R$, then a linear combination of $\{x,y,z,w \}$ is a finite sum of these things with coefficients from the set of scalars, here $\Bbb R$. So $\pi x + \sqrt{2} y$ is a linear combination since it is really $\pi x + \sqrt{2} y + 0z + 0w$. As you can see, my coefficients, which are scalars, come from $\Bbb R$.
If we decided we wanted our scalars to be $\Bbb Z$, the integers, then the above would not be a linear combination of elements in $\{x,y,z, w\}$ since some scalars are not integers. But an example of a linear combination with coefficients from $\Bbb Z$ is $19x + 5y - 3w$.
Now, we could choose our scalars to be wacky things, like the set of all $2 \times 2$ invertible matrices with real entries, for example. Then a linear combination of the elements of $\{x,y,z,w\}$ over this set of scalars could be $\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}x + \begin{bmatrix} 1 & 1 \\ 9 & 2 \end{bmatrix}z$.
Now, in your example, our coefficients, or scalars, happen to be integer polynomials. For example, in your example, a scalar would be $3x^{2} + 5x$, since this is a polynomial with integer coefficients, and the polynomial itself is acting as our coefficient. So, if we want to write a linear combination of $\{2, x \}$ with integer coefficients, we better write $oneThing * 2 + twoThing* x$ where $oneThing$ and $twoThing$ are coefficients (in this case, they better be polynomials with integer coefficients since these are our scalars). This is why $x^{2} + 2 = x \cdot x + 2$ is a linear combination. The $x$ has coefficient $x$, which is itself a polynomial with integer coefficients and thus a scalar.
If you now understand what a linear combination is above, the general way to write the definition of a linear combination of, say $\{ 2, x\}$ with scalars as polynomials with integer coefficients is to say "a linear combination is of the form $c_{1}2 + c_{2}x$ where $c_{1}$ and $c_{2}$ are scalars, i.e., polynomials with integer coefficients.
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