Transcript – Lecture 24b
OK, this is quiz review day. The quiz coming up on Wednesday will before this lecture
the quiz will be this hour one o'clock Wednesday in Walker, top floor of Walker,
closed book, all normal.
I wrote down what we've covered in this second part of the course, and actually I'm
impressed as I write it. so that's chapter four on orthogonality and you're
remembering these -- what this is suggesting, these are those columns are
orthonormal vectors, and then we call that matrix Q and the -- what's the key -- how
do we state the fact that those v- those columns are orthonormal in terms of Q, it
means that Q transpose Q is the identity.
So that's the matrix statement of the -- of the property that the columns are
orthonormal, the dot products are either one or zero, and then we computed the
projections onto lines and onto subspaces, and we used that to solve problems Ax=b
in -- in the least square sense, when there was no solution, we found the best
solution.
And then finally this Graham-Schmidt idea, which takes independent vectors and
lines them up, takes -- subtracts off the projections of the part you've already done,
so that the new part is orthogonal and so it takes a basis to an orthonormal basis.
And you -- those calculations involve square roots a lot because you're making
things unit vectors, but you should know that step. OK, for determinants, the three
big -- the big picture is the properties of the determinant, one to three d- properties
one, two and three, d- that define the determinant, and then four, five, six through
ten were consequences.
Then the big formula that has n factorial terms, half of them have plus signs and half
minus signs, and then the cofactor formula. So and which led us to a formula for the
inverse. And finally, just so you know what's covered in from chapter three, it's
section six point one and two, so that's the basic idea of eigenvalues and
eigenvectors, the equation for the eigenvalues, the mechanical step, this is really Ax
equal lambda x for all n eigenvectors at once, if we have n independent
eigenvectors, and then using that to compute powers of a matrix. So you notice the
differential equations not on this list, because that's six point three, that that's for
the third quiz. OK.
Shall I what I usually do for review is to take an old exam and just try to pick out
questions that are significant and write them quickly on the board, shall I -- shall I
proceed that way again? This -- this exam is really old.
November nineteen -- nineteen eighty-four, so that was before the Web existed. So
not only were the lectures not on the Web, nobody even had a Web page, my God.
OK, so can I nevertheless linear algebra was still as great as ever.