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[[[>50 This is not the only concept of "linear" - see further down the page. ]]] A straight line in two dimensional geometry has the characteristic that whenever you increase (or decrease) the variable $x$ by a particular amount, the variable $y$ will change by some multiple of that amount, no matter what the values actually are. So no matter what the values of $x$ and $y$ we have "change in y" is a constant times "change in x." If $y$ is 0 when $x$ is 0, that means we have: |>> $y~=~kx$ <<| In this case, $k$ is the ratio of changes. In general, when $x=0$ the vaue of $y$ will not necessarily be $0.$ As a result, a more general form is: |>> $y~=~kx+c$ <<| If there is a point, $(x_0,y_0)$ that the line goes through, that means that when $x=x_0$ we must have $y=y_0.$ As a result, we have this form: |>> $(y-y_0)~=~k(x-x_0)$ <<| Using algebra to rearrange that equation, we have: |>> $y=kx+(y_0-kx_0)$ <<| This is now in the same form as $y~=~kx+c.$ The constant $k$ in these equations is the "slope" of the line, and often the letter $m$ is used to represent the slope, rather than $k.$ !! Other definitions of "Linear" In the sense of "Linear Algebra" a function $f$ is said to be "linear" if it satisfies the following: * $f(x+y)~=~f(x)+f(y)$ * $f(kx)~=~k.f(x)$ A consequence of these conditions is that $f(0)$ must be $0.$ To see this, consider the first condition with $x=k$ and $y=-k.$ Linear functions don't need to be just from $\mathbb{R}$ to $\mathbb{R}$ or $\mathbb{C}$ to $\mathbb{C}$ but can be from any set to any set, provided it has suitable concepts of addition and multiplication - such a set forms a vector space. In this way we can then start to represent linear functions by matrices. This is why "Linear Algebra" is really the study of matrices, and matrix multiplication.