What is the meaning of the term "linear" [duplicate]

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$a_1x_1+a_2x_2+a_3x_3+...+a_nx_n=$ is called a linear equation because it represents the equation of a line in an n dimensional space. So "linear" comes from the word "line".Basically there should not be any higher power of x failing which the graph of the function will not be a straight line.

simillarly

$a(x)y+b(x)y'+c(x)y"+d(x)y'''+...+q(x)=0$ is also called linear differential equation because all the derivatives have power equal to 1 which is similar to the above definition of a linear equation.

A function f is called linear if:$f(x+y)=f(x)+f(y)$and$f(cx)=cf(x)$. Here c is a constant. In this definition of linearity of function "$f$" what does the word linear means? How does it relate to a straight line?

Finally what does the term linear means in case of linear vector spaces? Where is the reference to a straight line?

So, whether linear is just a word used in different contexts? Does it have different meaning in different situation? Or linearity refers to some relation to a straight line? At Least please explain how the linearity of function f and linear vector space relate to the equation of a line.

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1 Answer

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Linear functions are of the form

$$f(x) = ax$$

where solely the linear term is nonzero. The relation with straight lines is that if you see $x$ as a point in $\mathbb{R}^n$, then $x$, $f(x)$ and $0$ lie on a single line.

A function of that form will automatically satisfy

$$ f(x + y) = a(x + y) = ax + ay = f(x) + f(y)$$

and

$$ f(cx) = acx = cax = c f(x).$$

If, in a linear vector space, you draw a line between elements $A$ and $B$, then all points on that line, i.e. points of the form

$$ (1 - t) A + (t)B$$

will belong to the linear vector space. A consequence is that in a linear vector space, you can take $\textit{linear combinations}$ of elements, without leaving the space, i.e.

$$ A, B \in V \rightarrow c_1 A + c_2 B \in V \qquad \forall c_1, c_2 \in F$$

where $F$ is the field you used in defining the vector space.

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