| Summary of Ch. 5 Skills |
Ch 5 has quite a few inter-related skills. This page summarizes how to do them, and how to use them. It does not include applications beyond Ch 5, though we'll need some of these skills in Ch 6.4.
| Ch5 Skill | How to do it - | Use it for - |
| Finding an inner product | Use its definition. OR if there's an ONB of V, you can just find the scalar product (as in Rn). | Finding anything geometric - norms, projections, angles ... |
| Finding a basis of S^ | If S = R(A), then S^ = N(AT), a Ch 1 problem. |
Explaining the normal eqns. See also 5.6 - 7,8. |
| Least squares problems | Solve the normal equations using GE, or by factoring A = QR |
Real-life problems involving approximation, or overdetermined systems |
| Find the projection of v onto S | If S = R(A), it's the usual LS problem. Otherwise: a) find an ONB of S, b) find the projections of v onto these, c) add. [OR, set p = U UT v] |
1) approximation (eg of a function by
a polynomial, etc) |
| Find an ONB | Find a basis; then make it an ONB using GS. | With an ONB, we can use column vectors to find norms, inner products, projections, etc [as with the std basis of Rn]. |
| Factor A = QR | Apply GS to the columns of A to get Q. Certain numbers used go into R | 1) Simplifies the normal equations to Rx = QT
b 2) Gives an ONB for R(A). |
| Gram-Schmidt | Correct the direction of each vector by subtracting off its projection onto the current subspace. Then normalize. Include it and go on. | See A = QR above, but GS works even without column vectors. |
Some of the vocabulary (direct sums, the rank of A, etc) is mainly used in theorems and their proofs in this chapter.