How to Intuit if these are Linear Transformations or not ? [Strang P380, 7.1.3(c), (d)]

Question

On P376, Strang writes : "You'll get good at recognising which transformations are linear". In his video lectures, he does this; before algebra, he previses whether something's a linear transformation or not. Could someone please help with the following? Once again, I'm able to check with the definition of linear transformation; I'm not asking about algebraic verifications.

$P375.$ $T(\mathbf{v}) = \mathbf{Av + b}$ is the affine transformation. Since $T(c\mathbf{v} + d\mathbf{w}) \neq cT(\mathbf{v}) + dT(\mathbf{w}), $ thus this isn't a linear transformation.

Nonetheless, provided that $\mathbf{A, v, b}$ are scalars, then this is the function of a straight line with slope $a$ and $b$, which by definition is linear! Why this discrepancy? They're both called "linear"?

$P380. 3(c)$ $T(v_1, v_2) = v_1 - v_2$

$P380. 3(d)$ $T(v_1, v_2) = v_1v_2$

For these last two, I chose some scalars for $v_1, v_2$ and sketched but this didn't help.

Answer 1

For your first example, note that $T(0)\neq 0$. This violates a necessary property satisfied by all linear transformations.

For your second example, because it is a linear combination of the inputs, we see that it is linear.

Your third example involves multiplying two of the input values. Anything that does this will probably not be linear.

Here is a possibly useful perspective. It is true that all linear transformations are the same as multiplying by a matrix (after we fix some basis). So anything that cannot be matrix multiplication cannot be a linear transformation. Since we know that the zero vector multiplied by a matrix is zero, we see that (for example) $T(0)=0$ for all linear transformations. This is an easy way to see that transformations like your first example are not linear transformations.

Answer 2

With respect to the transformation $T(v) = Av + b$, it is an unfortunate coincidence that this is the equation of a line. This is not what it means to be a linear equation - instead, linearity means that changes in the input result in proportional changes in the output. In this context, we call equations of the sort $T(v) = Av + b$ to be an affine transformation. So yes, they are both called linear in different contexts. Both definitions are correct, but the idea of a linear transformation is perhaps the more important of two.

This is not the first time that some math notation has been abused, or that the same name has been given to different concepts, and nor will it be the last. (Perhaps this is due to us reusing many of the same metaphors over and over).

For the second and third questions, you should do a few more and see how your intuition grows. In particular, multiplying the arguments essentially never leads to linear transformations.