Fast procedure to know which transformations are linear

Question

I have the following transformations:

How can I tell if the transformation is linear?

Is there a fast way of knowing (by looking at equations)? or do I need to check through the theorems:

Thanks!

Answer 1

A way to use the theorems is this:

Let's take a look at D: $$\pmatrix{x_1 \\ x_2} \mapsto \pmatrix{0 \\ 9 x_2}$$

Let's check if it's a linear mapping! This is probably easiest if we just pick some numbers and try it out. Say we transform two vectors and we check for the first condition:

$$\pmatrix{2 \\ 3}+\pmatrix{4 \\ 1} \mapsto \pmatrix{0 \\ 36}=\pmatrix{0 \\ 27}+\pmatrix{0 \\ 9} \;\; \checkmark$$

What about the other condition? Let's again just pick some numbers:

$$4\pmatrix{1 \\ 2} \mapsto \pmatrix{0 \\ 72}=4\pmatrix{0 \\ 9} \;\; \checkmark$$

So D was linear!

Once you do this a couple of times, it will become much faster and you'll be able to quickly spot what a linear and a non-linear transformation looks like.

For instance, one that obviously fails is B, since it gives the square of one of the variables; if you double in the input, you quadruple the output, so that is not linear!

Answer 2

The right sides must all be (or simplify to) the form $(constant) \times (variable)$, or a sum of those. Of course the constant could be $1$ or $-1$, in which case you'll just see $(variable)$ or $-(variable)$. If you see a constant not multiplying a variable, or two variables multiplied together, or a variable to a power other than $1$, then it is not linear.