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PROFESSOR: The story begins with a statement that we will verify to some degree in the homework that is due on Wednesday of vector operators. This operator Sz, the one-line summary is that this operator is a vector operator under angular momentum-- under the total angular momentum. And as such, its matrix elements will behave like the matrix elements of Jz. So how is that true? What does it mean to say that S is a vector operator under J? It is to say that for J, S is like a vector. And that is a concrete statement that you should check whether it's true. The statement is that Ji Sj-- here, this i and j run from 1 to 3-- is equal to ih bar epsilon ijk Sk. That is the statement that S is a vector operator. You may have seen this in a previous course, in 805, where you might have proven that X and P are vector operators for L. And here, the proof is not all that difficult. In fact, it's almost obvious this is true, isn't it? Ji is Li plus Si. Li doesn't talk...

GILBERT STRANG: So today begins eigenvalues and eigenvectors. And the reason we want those, need those is to solve systems of linear equations. Systems meaning more than one equation, n equations. n equal 2 in the examples here. So eigenvalue is a number, eigenvector is a vector. They're both hiding in the matrix. Once we find them, we can use them. Let me show you the reason eigenvalues were created, invented, discovered was solving differential equations, which is our purpose. So why is now a vector-- so this is a system of equations. I'll do an example in a minute. A is a matrix. So we have n equations, n components of y. And A is an n by n matrix, n rows, n columns. Good. And now I can tell you right away where eigenvalues and eigenvectors pay off. They come into the solution. We look for solutions of that kind. When we had one equation, we looked for solutions just e to the st, and we found that number s. Now we have e to the lambda t-- we changed s to lambda...