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What is linear about a linear combination of things?. In linear algebra, the "things" we are dealing with are usually vectors and the linear combination gives the span of the vectors. Or it could be a linear combination of variables and functions. But why not just call it combination. Why is the term "linear" included?What is so "linear" about it?

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    $\begingroup$ If you replace all the vectors involved in the combination with a multiple of all of those vectors, you end up with a multiple of the combination. This scaling property is what makes it linear. $\endgroup$ Jun 29, 2018 at 6:03
  • $\begingroup$ @mathworker21 So would a linear combination of few linear functions result in some other linear function? $\endgroup$
    – GRANZER
    Jun 29, 2018 at 8:51
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    $\begingroup$ yes............. $\endgroup$ Jun 29, 2018 at 10:00

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The word "linear" has two distinct senses, one geometric and one algebraic. Linear combinations are linear in both senses, which is why the phrase is so apt.

Geometric Linearity

Let's take the geometric sense first, because that is the one that has not yet been explicitly mentioned by the other answers. If you studied Geometry in high school, you may (depending on the curriculum you followed) have learned some axioms for lines and planes. One of those axioms is:

If a plane $\textbf{P}$ contains two points $A$ and $B$, then it also contains the line $AB$.

This axiom expresses the intuitive notion of "flatness". Non-planar surfaces do not satisfy this property: for example of it you take two points on the surface of a sphere, the line joining those points does not lie on the surface; rather, it cuts through the interior of the sphere and exits the sphere. Inspired by this example, we might define the following property:

Definition. A subset $S$ of $\mathbb{R}^n$ is geometrically linear if $S$ contains all of the lines through the points of $S$.

Now let's take a few vectors and form their span, which is the set of combinations you can form by adding together scalar multiples of the vectors. If the vectors live in $\mathbb{R}^3$ (or more generally in $\mathbb R^n$) then that span is guaranteed to be "geometrically linear" in the sense described above. Whether that span is a plane, or a line, or some higher-dimensional analogue of those things depends on how many vectors you begin with and whether or not they are linearly independent, and that's a whole separate question; regardless, though, the span of any set of vectors is a subspace of the ambient vector space, and is "flat" geometrically. That's why we call them linear combinations.

Algebraic Linearity

In high school algebra, you study polynomial functions of a single variable, and the way the formulas for those functions relate to their graphs: $f(x)=ax+b$ determines a line, $g(x)=ax^2+bx+c$ determines a parabola, etc. More generally one can consider functions of more than one variable, like $f(x,y) = ax^2 + bxy + cy^2 + d$, or $g(x,y) = ax^3 + bx + cy^5$. By analogy with the single-variable case, we can make the following definition:

Definition. A a multivariable polynomial function is called algebraically linear if every term has degree $1$.

Note: This definition is actually stricter than the high-school level use of the word "linear", in the sense that a function like $f(x) = 3x +2$ would not be considered "algebraically linear", despite the fact that its graph is obviously a line, because the constant term has degree $0$. Some people are bothered by this mismatch of language; see https://matheducators.stackexchange.com/q/9835/29 for example.

In any case, with this definition established, an expression like $$ax + by + cz$$ would determine an algebraically linear function, whereas an expression like $$ax^2 + bxy + cz$$ would not.

Now it is a remarkable fact that these two notions of linearity -- the geometric and the algebraic -- coincide, at least in settings in which they are both meaningful. If the vectors in your vector space $V$ have a natural geometric interpretation -- for example if you think of $V=\mathbb R^2$ as a plane, or $V=\mathbb R^3$ as modeling he 3-dimensional world we live in -- then if you take any set of vectors, and form from it the set of all "algebraically linear" combinations, the span of the set is "geometrically linear".

What's nice about the algebraic formulation is that it also works in settings that are not easily interpreted geometrically. For example, if $f(x)$, $g(x)$, and $h(x)$ are any three functions on $\mathbb R$, you can form the set of "linear combinations" -- functions of the form $$af(x) + bg(x) +ch(x)$$ This describes a set of functions that can be "built from" $f, g, h$, in an algebraic sense, using only addition and scalar multiplication. A function built by multiplication, like $f(x)g(x)$, is not a linear combination; nor is a function built by composition, like $f(g(x))$. If the functions $f,g,h$ happen to be linearly independent, then you can think of their span as a 3-dimensional vector space -- and that vector space is geometrically linear as well, even though it may be difficult to visualize exactly what a "line" is. In fact, this is true even if the function $f,g,h$ themselves are "nonlinear". (Yes, you can build a linear combination of nonlinear functions; the result will in general also be a nonlinear function, but the set of all such combinations is a linear subspace.)

The notions of geometric and algebraic linearity go together so tightly that it is easy for people to forget that they are, to a novice, intrinsically different concepts: once you get used to this stuff, it's hard to remember that the notion of "line through the origin" has a geometric, intuitive meaning that precedes the formal notion of "$1$-dimensional vector subspace". This is an example of a phenomenon called the "expert blind spot"; I think it's particularly common when teaching linear algebra to people for whom the subject is new.

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It is linear because such a combination has the form

$$ (const_1) (quantity_1)+(const_2)(quantity_2)+\ldots $$

as opposed to expressions such as $$ (const)(quantity_1)(quantity_2) $$ or

$$ (const)(quantity)^3 $$

More generally, such forms preserve "linearity", in the sense that scaling by constants or adding or substracting them preserves the form.

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When we want to talk about things being linear we are restricting ourselves to only two potential operations

  1. We can add things together
  2. We can scale things

We call these operations linear because, well, they operate on a line! When you scale a vector, the new vector is a "further out" version of the original on the same line. When we add two vectors together, we are scaling and changing the vector angle a bit.

Notice that nothing is curvy!

So when we talk about a linear combination, we say that you can create a combination from a set of vectors $\{v_1,v_2,...v_n\}$ using only those two operators

$$ v_{new} = c_1v_1 + c_2v_2 + ... c_nv_n $$

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  • $\begingroup$ Then what operation would make it curvy? $\endgroup$
    – GRANZER
    Jun 29, 2018 at 8:50
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    $\begingroup$ @GRANZER Squaring for example $x^2$ graph is curvy vs. $2x$ graph. $\endgroup$
    – wjmccann
    Jun 29, 2018 at 14:48

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