If $T(x) = Ax + b$, is $T$ a linear transformation? I don't quite understand the relation between $T$ and $A$, is $T$ just $A$ with an added or removed dimension? I know the qualifications for having a transformation to be linear where $T(u+v) = T(u) + T(v)$ & $T(cu) = cT(u)$, but I don't see where the $A$ comes into play.
2026-04-06 19:27:07.1775503627
linear transformation qualifications
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Only if $b=0$. Otherwise $T$ is affine.
It should be clear from the properties of matrix multiplication that $T(x) = Ax$ is linear (i.e. it satisfies both of those properties) for any matrix $A$.
But this is only one type of linear transformation. An important type, granted, but still there are other linear transformations. For example, $I(f) = \int_0^1 f(x)\,dx$ is another linear transformation. If you've taken calculus, this should also be easy to prove.