trying to be a mile wide AND a mile deep
At the center of almost any proof involving linear transformations is to apply linearity to move between the domain and codomain, preserving the structure of linear combinations. Statements proved in such a way: all linear transformations take 0 to 0 images of subspaces are subspaces (special example: image) preimages of subspaces are subspaces (special example: […]
I like examples. diagonalizable not diagonalizable invertible I_n, rotation about non-axis line by, say, 60 degrees in R^3, rotation 90 degrees in R^2: not invertible zero matrix, rotation 90 degrees plus projection onto yz-plane in R^3:
I didn’t really like the cofactor construction of the inverse of a matrix with nonzero determinant, to prove said inverse exists. I’m willing to accept the equation det(MN) = det(M)det(N) on faith, since I am confident I could work that out if I really had to. With that in my pocket, here’s an explanation of […]