# Multiple inputs and linearity?

I'm struggling to understand how the ratio of inputs to outputs relates to linearity.

Determine if the function is linear or nonlinear.

If I am not mistake, this has 2 inputs, 1 output: $f(x) = 2x_1 + 3x_2$

But this function has 2 inputs, 2 outputs: $f(x) = [3x_1 + 2x_2, -4x_1]$

I tried superposition and homogeneity but couldn't get anything sensible.

Apart from anything else, "the number of inputs" or outputs is not a clearly defined concept. If you have, as in your second example, $$f(x_1,x_2)=(3x_1+2x_2,\,-4x_1)\ ,$$ you could say that $f$ has two (numerical) inputs, or one (vector) input. It's a matter of language, not really a question of mathematics. Another example: is $$\pmatrix{1&2\cr3&4\cr}$$ four numbers, or is it one matrix? (Or two column vectors maybe?)