405 Summary

Summary of lectures and homework assignments

Here is what we have done so far:

8/30: We talked about fields. Most of linear algebra can be done using scalars in any field (a notable exception being inner products which require for example a notion of positivity). That being said, after this mention of other fields to broaden your horizons, we shall concentrate on the fields R (the real numbers) and C (the complex numbers). We also discussed the algebraic rules satisfied by matrices. They are almost the same as for fields except you don't have AB=BA and A^-1 does not exist for some nonzero matrices A. I handed out a test of some concepts from first semester linear algebra which I want you to hand in on 9/6. The material I covered roughly corresponds to chapter 1 of H&K.
9/1: 2.1,2.2 We talked about vector spaces and subspaces. Since H&K does not give a notation for the set of functions from S to F we use the standard notation F^S for this. Probably every vector space we encounter will be a subspace of F^S for an appropriate S. Homework due 9/8:

2.1: 5, 6, 7
2.2: 1, 2, 5, 7
Grade distribution, (scaled down by a factor of 7): 2, 2.5, 3, 5.4, 6.3, 6.5, 7, 7, 7.1, 7.5, 7.5, 7.7, 8.2, 8.7, 8.7, 9.5, 9.7

9/6: We covered more material from 2.1,2.2. We showed that c0 =0, 0a = 0, and (-1)a = -a. We talked about linear combinations of vectors, showed the solutions to AX=0 form a subspace of F^nxk, talked about the subspace spanned by a set of vectors.
9/8: We talked a little about the homework. We looked at the sum of subspaces W1+W2+...Wk. We started on 2.3, linear dependence and independence. I handed back the diagnostic exam. The score distribution was:

90-100: 11
80-89: 4
70-79: 3
50-69: 1
40-49: 4

9/11: We continued with section 2.3. We defined basis and gave some examples. We showed that any two bases have the same number of elements. Homework due 9/15:

2.3: 2, 3, 6, 7, 11

9/13: We finished 2.3, showing that any linearly independent subset of a subspace W of a finite dimensional vector space V can be completed to a (finite) basis of W. In particular, any subspace of a finite dimensional vector space has a finite basis. We showed that if W1 and W2 are two subspaces of a finite dimensional vector space then dim(W1)+dim(W2) = dim(W1+W2) + dim(W1 intersect W2).
9/15: We covered 2.4, coordinates with respect to a basis, and talked some about the ideas in 2.5. Our goal in 2.5 was to show that the row reduced echelon form of a matrix is unique, so if you do two different sets of row operations they will still end up with the same rref. In fact we sketched why the rref of A is determined by the row space. So in fact the rref depends only on the row space.
9/18: We started talking about 3.1, linear transformations, giving some examples, showing a linear transformation is uniquely determined by its values on a finite basis, and showing the range and kernel (or null space as H&K calls it) is a subspace. Homework due 9/22:

3.1: 1(and say why), 2, 3, 4 (give reason), 8, 9, 12
3.2: 1, 6

9/20: We talked about most of 3.2. We showed that the set of linear transformations from V to W forms a vector space L(V,W). We showed the composition of linear transformations is a linear transformation. We showed that the inverse of a linear transformation, if it exists, is a linear transformation. We gave an example of an onto linear transformation from V to V which is not one to one, and a one to one linear transformation from V to V which is not onto. This is not possible in finite dimensions.
9/22: We proved Thm 2 of 3.1 and talked about isomorphisms, showing that finite dimensional vector spaces are isomorphic if and only if they have the same dimension.
9/25: We talked about representing a linear transformation as a matrix, 3.4. HW due Monday, 10/2.

3.3: 3, 6, 7
3.4: 1, 8, 10
3.5: 1, 2, 7

9/27: We talked about linear functionals, dual bases, anihilator, 3.5.
9/29: We talked about the double dual and showed there is a natural isomorphism between V and V** if V is finite dimensional. In infinite dimensions, we showed the natural map is one to one (assuming the axiom of choice) but is not an isomorphism. 3.6.
10/2: We talked about the transpose of a linear transformation, 3.7
10/4: No new material, we did some problems from 3.6 and 3.7.
10/6: Exam #1
10/9: Ideals of polynomials, 4.4. An ideal in F[x] is a subspace J of F[x] which is closed under multiplication by x. Equivalently, it is closed under multiplication by any polynomial in F[x]. It turns out all ideals in F[x] are principal, that is, any ideal is the set of all multiples of some fixed polynomial p(x). Homework due 10/16:

4.2: 1
4.4: 1, 4
6.2: 1, 5, 11, 15

10/11: 6.1, 6.2, 6.3 We talked about the annihilating ideal of a linear transformation T:V->V, the set of all polynomials p so that p(T)=0. The minimal polynomial of T is the monic polynomial which generates this annihilationg ideal. We then started talking about characteristic values and characteristic vectors, which are just other names for what you probably learned as eigenvalues and eigenvectors. The due date for homework has been changed to Monday.
10/13: More 6.2 and 6.3. We showed that any characteristic value of T is a root of its minimal polynomial and vice versa. We then defined the characteristic polynomial of T and stated the Cayley Hamilton theorem that the characteristic polynomial annihilates T. I find the proof of this given in H&K 6.3 rather obscure so let's wait for later to prove it. Meanwhile I gave a sort of proof, showing that it is true for diagonalizable operators and then noting that char poly(T) is a continuous function on C^nxn which is 0 on the diagonalizable matrices and any matrix may be arbitrarily closely approximated by a diagonalizable matrix, hence by continuity this function is 0 on all of C^nxn.
10/16: We started talking about invariant subspaces, 6.4. HW due 10/23:

6.3: 2, 4 (you may assume prob 3), 6
6.4: 1, 3, prove 5, 9

10/18: We talked about the T conductor. Then proved that a linear operator is triangulable if and only if its minimal polynomial is a product of linear factors. We started proving that a linear operator is diagonalizable if and only if its minimal polynomial is a product of distinct linear factors.
10/20: We finished the proof that a linear operator is diagonalizable if and only if its minimal polynomial is a product of distinct linear factors. We stated the result that a family of commuting diagonalizable operators is simultaneously diagonalizable and that a family of commuting triangulable operators is simultaneously triangulable. We got off on a tangent though and didn't prove it. the proof is in the book in section 6.5.
10/23: 6.6, 6.7, We talked about the (internal) direct sum of subspaces. If V is the direct sum of T invariant subspaces, then equivalently we may find a basis which puts T in block diagonal form. HW due 10/30:

6.5: 1a, 2
6.6: 1, 4, 5
6.7: 2, 6 (but he means example 5 of section 6.3)
6.8: 1

10/25: 6.7 We talked about the projection operators E_i associated to a direct sum decomposition. I also gave a proof of the Cayley-Hamilton theorem if the field F is the complex numbers (and in fact the proof works for any algebraically closed field if you know what that is). It used the fact that any operator on a finite dimensional vector space over C is triangulable.
10/27: 6.8 We proved an important special case of the Primary Decomposition Theorem, simplifying things a little by assuming that the minimal polynomial is a product of linear factors (as would be true if the field were the complex numbers). This simplification does not change the flavor of the proof at all, but does mean we don't need to spend time talking about prime polynomials. Anyway, when reading Theorem 12 in the text you can assume for our purposes that p_i = x-c_i.
10/30: I talked about a simplified version of 7.1, 7.2 which covers what we need for the Jordan form. Notes on this are here. HW due 11/8:

7.1: 1, 5
7.2: 1
7.3: 1, 3, 4, 5, 8, 10, 14

11/1: I talked a little more on the simplified 7.1, 7.2.
11/3: 7.3 I talked about getting a matrix into Jordan form. After class I postponed the homework due date to Wednesday.
11/6: 7.3 more on the handout from 10/30. Here are a few notes on Jordan form, they may help with the homework.
11/8: We did a number of homework problems. One was on square roots and I made a few comments on finding the square root of a linear operator.
11/10: We began talking about inner products. Note that we are specializing to the real or complex fields only. After class someone floated the idea of delaying the exam until Monday 11/20. We'll talk about this next time.
11/13: We discussed the Gram-Schmidt process and did some problems. Exam #2 is definitely delayed until Monday, 11/20. But anyone wishing to take an alternate exam on Friday 11/17 may do so. It will be given after class. HW due 11/20:

8.1: 1, 2, 6, 9, 12
8.2: 3, 6, 7, 10, 17

11/15: We showed the Cauchy-Schwartz inequality, the triangular inequality, and the Pythagorean theorem. We also looked at the orthogonal complement of a set and showed that in finite dimensions that the orthogonal complement of the orthogonal complement of S is the span of S. This not true in infinite dimensions, for example if V is complex valued continuous functions on [0,1] and $ is the set of e^{2k\pi i t} for k=0, 1, -1, 2, -2, ... then the orthogonal complement of S is 0. Thus S perp perp = V which is not the span of S.
11/17: We looked at the many uses of *, as the conjugate transpose, as the adjoint, and as the dual. These are all related to each other and any ambiguities can be resolved.
- First the dual. If T:V->W is linear then there is an induced linear transformation T*:W*->V* which is given by T*f=fT, in other words for each a in V, T*f(a) = f(Ta).
- Next the adjoint. If T:V->W where V and W are inner product spaces, then the adjoint of T is a linear transformation T*:W->V so that (Ta | b) = (a | T*b) for all a in V and b in W. (H&K only considered the case W=V but this is not needed).
- Finally the conjugate transpose. If A is an nxm matrix over R or C then A* is an mxn matrix.

^mx1

^nx1

11/20: exam
11/22: 8.4 We talked about isomorphisms of inner product spaces and unitary matrices.
11/27: More 8.4, but I'm doing some things differently. First using the Gram-Schmidt process on its columns, any matrix A can be written A=QR where Q is unitary and R is upper triangular. H&K does a variation on this in Theorem 14 by essentially working with the transpose, but the QR decomposition is more standard. Now if B is any triangulable matrix, there is a matrix P so that T = P^-1BP is upper triangular. Let P=QR be the QR decompostion of P. Then Q^-1BQ = RTR^-1. But R, T, and R^-1 are all upper triangular, so Q^-1BQ is upper triangular. So any triangulable matrix is triangulable using a unitary matrix Q. The corresponding statement for operators is that if T:V->V is an operator on a finite dimensional inner product space, then there is an orthonormal basis B={b1,b2,...,bn} so that [T]_B is upper triangular. The proof is to take any basis A so [T]_A is upper triangular, then do Gram -Schmidt on A to get B, and then [T]_B will automatically be upper triangular also. Last HW due 12/4

8.3: 1, 4, 6, 9
8.4: 1, 2, 6, 10
8.5: 1, 4, 8, 9

11/29: 8.5: We showed that any normal operator T:V->V on a complex inner product space is orthonormally diagonalizable, i.e., there is an orthonormal basis B of V so [T]_B is diagonal. The proof I gave was different from that in H&K, As we did on 11/27, you can pick an orthonormal basis B so that [T]_B is upper triangular. But then we showed that any upper triangular matrix which is also normal must be diagonal, so [T]_Bis diagonal. In the real case we showed that an operator is orthonormally diagonalizable if and only if it is symmetric. We showed a symmetric operator is orthonormally diagonalizable. The trick is to represent it by a symmetric matrix which we then temporarily think of as being complex Hermitian matrix when we can then conclude that all its characteristic values are real, so its minimal polynomial is a product of linear factors, so it is triangulable, so it is orthonormally triangulable, but any upper triangular symmetric matrix must be diagonal, so it is othonormally diagonalizable. We started out looking at what you do with normal real matrices which are not symmetric, such as skew symmetric and orthogonal matrices, but didn't get far.
12/1: We looked at real skew symmetric operatorss and showed that there is an orthonormal basis so that the matrix of the operator is block diagonal with either 1x1 blocks of 0 or 2x2 blocks [0 b; -b 0]. I'll write out some notes on this stuff since it is not in the book.
12/4: We talked more about real normal matrices.
12/6: We did a matlab demo on real normal matrices and talked about a real Jordan form for any real n x n matrix. Preliminary writeup is here.
12/8: Review
12/11: Review
12/13: I'll drop by our classroom at 1:00 just in case anyone has any questions.