Section 5.3

I understood the Krylov subspace method with the reference to the QR factorization. However, I still wonder how these algorithms compare to each other with time. I was a little confused in Example 5.6. In the comments it said that the vector was not yet orthogonal to the first direction and I was a little confused about what it meant to be orthogonal to the first direction and also what it was meant in the comments when the program continued running in different directions.