13th
Linear Algebra
For those of you playing along at home, I’m now trying to work on a dimensional analysis problem, which are supposed to be easy but which I didn’t understand when we did them on the last homework. I think it’s because my linear algebra is rusty. So here are some linear algebra facts:
You can think of a matrix as a function that takes in a vector and outputs a different vector. Because when you multiply a vector by a matrix, you get a vector (wikipedia will tell you how!). Like a rotation matrix in three dimensions or something, I multiply my point in 3 dimensional space (which counts as a vector) by some 3 x 3 matrix, and I get a different point in 3 space. If I did this to every point in say, a 3-d image, and I really had a rotation matrix, the result would be the same image rotated into a different orientation.
Now when you have a function like that, also often called a map, it’s kernel is the set of everything that the function sends to 0. Like if f(x,y,z) = (0,0,0), then (x,y,z) is in the kernel. So the kernel of a matrix is the set of all vectors that are “sent” to 0 by the matrix. I.e. if v is a vector and M a matrix and Mv = 0*, then v is in the kernel of M. The kernel is also called the nullspace.
Now I’m trying to find the nullspace of a matrix formed by expressing the dimensions of quantities in terms of mass, length, and time, because the way my matrix is set up, the things this matrix sends to 0 will be dimensionless quantities. There’s something more I need to remember about free variables before I can do this. One-woman math squad away!
*here 0 is a vector all of whose entries are 0
