| Week 1 | January 17, 19 |
| Tu in class we did problems (1.1) 27, 37 and (1.2) 29. 61 | HW1 : (1.1) 2, 4, 7, 29 due Th 19 |
| Th we did matrices in RREF and elementary row operations (resp. top and bottom of page 16) and two ways to evaluate a product Ax (top page 29 and bottom page 30) | HW2 : (1.1) 18, 23 (1.2) 2, 4, 18, 22 (1.3) 14, 36, 58 due Th 26 |
Week 2 | Janunary 24, 26 |
| Tu we did problems (1.1) 17 (1.2) 46 (1.3) 37 and defined linear transformations (Defn 2.1.1 and Thm 2.1.3) | |
| Th we pointed out the connection between composition of linear transformations and matrix products (L after T has matrix BA - Defn 2.3.1 / Thm 2.3.4) and we used row reduction techniques from Chapter 1 to find the inverse of a matrix (pages 81 and 82) | HW3 : (2.1) 6, 8, 44, 52 (2.3) 4, 20 (2.4) 12, 18 due Th 2 |
Week 3 | February 31, 2 |
| Tu we showed that projections on a line or a plane are linear transformations, we found a short formula for projection on a line (x goes to kw where k = (x.w)/(w.w)), and we derived in three different ways the matrix A for a projection (for this part use notes from class rather than section (2.2) from the book). This completes Chapter 2 | Notes from class |
| Th we went through sections (3.1), (3.2), including important definitions: the image im(T) and kernel ker(T), linear combinations and span, linearly independent / linearly dependent and basis, and the Theorems 3.1.3, 3.1.4, 3.1.6, 3.2.2 | HW4 : (3.1) 10, 16, 30, 34, 42, 50, 52 (3.2) 2, 6, 8, 18, 32 due Th 9 |
Week 4 | February 7, 9 |
| Tu Section (3.3) dimension of a vector space, how to find the image and kernel for a given matrix A | |
| Th Section (3.4) matrix of a linear transformation wrt a given basis | HW5 : (3.3) 2, 6, 24, 26, 30, 38 (3.4) 8, 10, 26, 40, 56 due Th 16
|
Week 5 | February 14, 16 |
| Tu Sections (4.1) (4.2) Definition of linear spaces and examples (R^n, P_2, R^{2x2}), linear transformations T: V --> W | |
| Th Section (4.3) Matrices A and B of a linear transformation wrt different bases (a1,...,an) and (b1,...,bn) and their relation (AS = SB) | Example |
Week 6 | February 21, 23 |
| FIRST MIDTERM FEB 21 (IN CLASS) | Exam1-test |
| Ch.5 (5.1) Definitions: inner product, orthogonal, norm. Theorem: Cauchy-Schwartz inequality, we gave two different proofs | HW6 : (5.1) 12, 40-46 (5.2) 8, 22 |
Week 7 | February 28, 1 |
| Ch.5 (5.2) Gram-Schmidt Process, QR Factorization | |
| Ch.5 Projection on span(v1,...,vn) (the book considers
in Theorem 5.1.5 the special case that v1,...,vn are orthonormal, in class we discussed the general case where v1,...,vn are arbitrary vectors), (5.3) A matrix A is orthogonal if A^T A = I, (5.4) Defn 5.4.4, Thms 5.4.5 - 5.4.6 - 5.4.7 (in class we discussed the relation between Ax=b and the projection of b on im A) | HW7 |
Week 8 | March 6, 8 |
| Ch.5 (5.4) A(A^T A)^-1 A^T becomes Q Q^T for A=QR with Q orthogonal | |
| Ch.6 (6.2) The determinant is the unique map A -> det(A) that maps I -> 1 and such that after (1) multiplication of a row of A by k (2) swapping of two rows or (3) adding a multiple of a row to another row det(A) becomes (1) k det(A) (2) - det(A) or (3) det(A), respectively. From these properties we immediately obtain : (6.2.4) det(A) not 0 if and only if A is invertible, (6.2.10) the cofactor expansion of the determinant, (6.3.8) Cramer's rule and (6.3.9) the inverse of a matrix A^{-1} = adj(A) / det(A) | HW8: (6.2) 6, 12, 14, 18, 26, 30, 50 (6.3) 22, 30 |
Week 9 | March 13, 15 |
| Ch.6 Computation of A^{-1} in three steps: matrix M of minors, cofactor matrix C and adjoint matrix adj(A). Start of Ch.7: Coyotes and roadrunners example, Cases 1, 2, 3, Figure 6 | |
| Ch.7 (7.1) Defn 7.1.1 - Eigenvectors and eigenvalues. (7.2) Thms 7.2.1, 7.2.5 - Characteristic equation, Characteristic polynomial, Thm 7.2.2 - Eigenvalues of a triangular matrix, Defn 7.2.6 - Algebraic multiplicity of an eigenvalue, (7.3) Defns 7.3.1, 7.3.2, 7.3.3 - Eigenspace, Geometric multiplicity of an eigenvalue, Eigenbasis, Thm 7.3.5 - A matrix with n distinct eigenvalues has an eigenbasis. If the columns of S form an eigenbasis for A then AS = SB for a diagonal matrix B (as in Section 3.4, page 145) | HW9: (7.1) 6, 8, 10, 34, 46, 50 (7.2) 38 |
Week 10 | March 20, 22 |
| Spring Break | |
| Spring Break | |
Week 11 | March 27, 29 |
| Ch.5 (5.5) Using defn 5.5.1 (Inner product for vector spaces other than R^n) most of the material in previous sections generalizes: norm, orthogonality, orthogonal projection, and least square approximation. Important vector spaces: C[a,b], continuous functions on the interval [a,b] with inner product
given by the integral of f(x)g(x) from x=a to x=b. | |
| Ch.7 (7.4) A is diagonalizable if it is similar to a diagonal matrix (if S^{-1}AS=D for some S) | |
Week 12 | April 3, 5 |
| SECOND MIDTERM APR 3 (IN CLASS) | Exam2-test |
| Ch.7 (7.5) Dynamical systems with complex eigenvalues, glucose-insulin example, back to (2.2) - Interpretation of the 2x2 matrix [[a,-b],[b,a]] as rotation followed by scaling (thm 2.2.4) | HW10: (7.3) 36 (7.4) 32, 36 (7.5) 14 (hint: thm 7.5.3) - accepted till Monday 4/16 when handed in before 11:30 in the box outside my office |
Week 13 | April 10, 12 |
| Ch.7 (7.6) Dynamical systems with complex eigenvalues (thm 7.6.3) | |
| No class | HW11: (9.1) 26, 30 (9.2) 22-26, 32 - due 4/19 in class, or not later than 11:30 am 4/20 in the box outside my office |
Week 14 | April 17, 19 |
| Ch.9 (9.1) Continuous dynamical systems, case of real eigenvalues (thm 9.1.3, compare with them 7.1.3) | |
| Ch.9 (9.2) id., case of complex eigenvalues (thm 9.2.3, compare with thm 7.6.3) | HW12: (8.1) 6, 8, 10 (8.3) 14 - due 4/26 in
class, or not later than 11:30 am 4/27 in the box outside my office |
Week 15 | April 24, 26 |
| Ch.8 (8.1) Properties of symmetric matrices (thms 8.1.1, 8.1.2, 8.1.3) (8.2) Quadratic form, principal axes (defns 8.2.1, 8.2.6) | |
| Ch.8 (8.3) Singular value decomposition (thm 8.3.5) | |
Week 16 | May 31-4 |
| Last day of class Tuesday May 1 | |
| Question hours (tentatively): Friday May 4 at 4, Sunday May 6 at 2, Wednesday May 9 at 4 | |
Week 17 | May 7-11 |
| Final Exam P13 (TR 11:00-12:20) Monday May 7, 1:30 - 4:30 pm | |
| Final Exam Q13 (TR 12:30-1:50) Thursday May 10, 8:00 - 11:00 am | |