Lecture 16: Geometric examples of Linear Transformations (Nicholson Section 2.6/Section 4.4)

Alternate Video Access via MyMedia | Video Duration: 48:48

Description: Started by reviewing definition of linear transformations. Review of the “before” and “after” presentation of what a transformation from R^2 to R^2 does. (There’s nothing special about this before and after way of presenting what a transformation does --- it’s just that it’s easiest to draw when the domain and codomain are in R^2 because I can draw everything easily rather than trying to draw things in R^3 or R^5 or something…)

4:25 --- In terms of this “before” and “after” presentation, what does it mean for T(x+ y) = T(x) + T(y)?

7:49 ---  In terms of this “before” and “after” presentation, what does it mean for T(r x) = r T(x)?

8:43 --- Is the mapping T([x1;x2]) = [1;x2] a linear transformation? We can see pictorially that it isn’t. Separately, we can check that T([2;0]+[0;2]) doesn’t equal T([2;0]) + T([0;2]). (To show that something isn’t a linear transformation you just have to give a single example where it breaks one of the rules.)

11:56 --- started geometric transformations. First example: dilation. Proved it’s a linear transformation.

19:55 Can I represent dilation as a matrix mapping? Yes, but need to be careful --- what matrix you get depends on what basis you use for the domain. (One thing that’s confusing/important is that when I defined dilation, it was done w/o referring to any specific basis --- it was defined simply as “given a vector in R^2, double its length”. I didn’t need to refer to the coordinates of the vector --- the moment I refer to the coordinates of the vector I’ve implicitly chosen a basis. For example, if I’m in matlab and I write “x =[2;3]” then implicit in this is the standard basis and what I mean is “x = 2 [1;0] + 3 [0;1] = 2 e1 + 3 e2”.)

32:08 --- Second example: “silly putty” transformation. Note: represented this transformation requires some sort of coordinates because it does one thing in one direction and nothing in another. And so I define it directly in terms of coordinates (I’ve implicitly chosen a basis like {[1;0],[0;1]}). Checked that the transformation is linear and represented it as a matrix transformation.

40:10 --- Third example: shear transformation. Represented it using coordinates (you can check on your own that it’s a linear transformation) and represented it as a matrix transformation.