Confusion about properties of linear transformation.

Question

I learned from 3Blue1Brown's Linear Algebra videos that one of the rules for a transformation to be linear is that the origin remains fixed in place after transformation. So if $T(x) = x+a$ is a given transformation, I know that by inserting $x=0,$ it is not a linear transformation. I am not able to understand what is non-linear about it.

Answer 1

If $a\not=0$, then $T(0)\not=0$. The transformation is what we call affine, but not strictly linear.

Answer 2

It's not a linear transformation on $\mathbb{R}$ viewed as a vector space precisely for the reason you state. It is reasonably and often called a linear function from the real line to itself because it's a polynomial of degree $1$ (a linear polynomial) and its graph is a straight line.

It's quite reasonable to be confused by this when you first see it. You won't be from now on. You can always tell from context what "linear" means there.

Answer 3

Because $T(0+0)=a$, whereas $T(0)+T(0)=2a$, which is different from $a$ (unless $a=0$, of course). But one should have $T(0+0)=T(0)+T(0)$ if $T$ was linear.