trying to be a mile wide AND a mile deep
1. There is no one multiplication for vectors. You can define multiplication-like operations; some give scalars (dot product and other inner products) and some give vectors (cross product). Nicely, these behave like regular products when it comes to vector-valued functions: the product rule applies when you differentiate (though you must maintain ordering with cross product!). […]
Linear independence, spanning, and the three bears Papa’s bed is too hard – linearly independent, but too spare to span. Mama’s bed is too soft – spans, but too many options to suffocate in. Baby’s bed is just right – enough to span and nothing more. A basis! Coordinate vectors and sweets A restaurant serves […]
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:
Here are slide presentations for a linear algebra class that claims to include applications, but really barely has enough time to cover the nonapplied material. I made two presentations that could be done in 20 minutes of class time. These are both incorporated to some degree into the math club linear algebra talk I gave, […]
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 […]