I like how the professor kept the definition of linear transforms simple during lecture, but I personally just prefer to have a formal definition. For anyone who may also prefer as such or maybe needs a refresher on the topic, a linear transform is defined as "a function T:R^n→R^m that satisfies the following properties:
T(x+y)=T(x)+T(y)
T(ax)=aT(x)
for any vectors x,y ∈ R^n and any scalar a ∈ R."
I think it's also worth noting that all matrix transformations are linear transforms, but the vice versa is not necessarily true.
llejj
To add on to the comment above, about how not all linear transformations can be represented by matrices. For example, the map from a function to its derivative is linear, but in some sense it is infinite dimensional so you can't write it as a matrix.
I like how the professor kept the definition of linear transforms simple during lecture, but I personally just prefer to have a formal definition. For anyone who may also prefer as such or maybe needs a refresher on the topic, a linear transform is defined as "a function T:R^n→R^m that satisfies the following properties: T(x+y)=T(x)+T(y) T(ax)=aT(x) for any vectors x,y ∈ R^n and any scalar a ∈ R." I think it's also worth noting that all matrix transformations are linear transforms, but the vice versa is not necessarily true.
To add on to the comment above, about how not all linear transformations can be represented by matrices. For example, the map from a function to its derivative is linear, but in some sense it is infinite dimensional so you can't write it as a matrix.