Explain the terms 'linear' and 'non linear'?

Answer 1

There are many ways to explain and define this. I will not repeat what is found in book or the web, rather explain an easy way to think of linear and non linear. Think of a linear map as one that maps straight lines to straight lines. It must map the origin to itself. Non linear maps do NOT do this. Look at f(x)=x, this maps every point including the origin to itself. So if I have 3 points on a line, then f will map them to a line. And it will map the origin to itself. So f is linear. f(x)=3x will do the same thing. In linear algebra we say f is linear if f(ax)=af(x) and f(x+y)=f(x)+f(y) This is really the same as what I said above. f(x)=x^2+2 will not be linear. If you take a line, it will not be mapped to a line. Furthermore f(0)=2 so it is NOT linear. Here is one more example that may help to see what's going on. Let's define the map T as T(x,y)=(x+y ,y) This is called a shear. What does it do to lines and the origin? T (0,0)=(0,0) so the origin is mapped to itself as we need for linearity. Let us look at the y axis which is x=0 T(0,y)=(y,y) So all the points on the y axis such as (0,1),) (0,2).. are mapped to (1,1), (2,2)... Which is a line ( it is the line y=x) So this does maps lines to lines. Now f(aX) just mean multiply each point by a, so f(a(0,1)=f(0,a)=(a,a) and a(f(0,1))=a(1,1)=(a,a) this is also required for linearity. Last let X=(0,1), y=(0,2) f(x)+f(y)=(1,1)+(2,2)=(3,3) and f(0,1+2)=f(0,3)=(3,3) This is not a proof since we used specific examples and numbers, however it is pretty easy to mimic what was done with arbitrary variables.