Lecture 1 (Aug 25): Givental's book, section 1: Vector.
HW 1: page 7 in [LA], pick at least 4 problems from 1-12 and solve it. (Givental's exercises are too interesting to be put in the quiz. So our quiz will be decoupled from the HW actually. Our first quiz will be a test run, and no grade will be recorded.)
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Lecture 2 (Aug 30): section 2: analytic geometry: conic curves, linear transformations.
Lecture 3 (Sep 1): review. affine space and affine linear transformation. Set theoretic notation. (ODE section 1.3)
HW 2: [ODE] (conic curve:)p11, read the Example. (matrices:)Ex 1.3.1 (b,c,d,g). Ex 1.3.2 (a-f)
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Lecture 4 (Sep 6): [ODE 1.3]. 2×2 matrices: determinant, inverse. Orthogonal transformation.
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Lecture 5 (Sep 8): [LA] Ch 1 Sec 3. Orthogonal Transformations. Complex Numbers.
video HW 3:
[ODE]1.3.4, (The notion of similar matrices are on top of page 19)
[LA] p22, 44,45,47,48,49,51,54,57,58.
quiz will be about complex numbers, similar to [LA] exercise above.
Lecture 6 (Sep 13) [LA] 1.4: Four theorems in Linear Algebra (intro)
video Lecture 7 (Sep 15) [LA] 2.1 Matrices, 2.2 Determinants
audio only HW 4: : 86, 88, 91, 95, 98, 99, 100, 107, 111, 112, 114, 116, 117, 119
Lecture 8 (Sep 20) [LA] 2.2 Determinants. 5 properties of det(A), $\det(AB)= \det(A) \det(B)$.
video Lecture 9 (Sep 22) [LA] finish up 2.2 about cofactor and block matrix multiplication
video HW 5: 110, 118, 132, 135, 138, 139, 140, 143
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Lecture 9 (Sep 27) [LA] 2.3 Abstract Vector spaces. Fields.
video Lecture 10 (Sep 29) [LA] 3.1 Dimension and Ranks
video HW 6: 150,151,152,155,159(notice $W^\perp$ in general lives in the dual space),160, 165, 167
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Lecture 11 (Oct 4) [LA] 3.1. rank theorem
video Midterm 1: Oct 5. 11:10-12 , one page (both side) cheat-sheet allowed
Lecture 12 (Oct 6) [LA] 3.1 (still rank theorem)
video HW 7: [LA] 178, 179, 180, 184, 185, 187
Give an exposition of the Remark in page 79 (the one after Corollary 1)
195 (this one is a bit hard, try a few examples first)
Lecture 13 (Oct 11) [LA] 3.2 Gauss Elimination. Echelon form
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Lecture 14 (Oct 13) [LA] Q&A on dual vector space, dual basis and quotient space.
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HW 8:
[LA] read about LPU decomposition. give some 2×2 and 3×3 examples where the permutation matrices is not the identity. read about flag manifold
197, 199, 201, 202, 206*(flag variety), 208*. *=extra
Lecture 15 (Oct 18) [LA] 3.2 LPU decomposition. Flag variety
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Lecture 16 (Oct 20) [LA] 3.3 Quadratic form
video HW 9: (1) [LA] 211 - 219. (2) Turn the proof of Lemma of existence of orthogonal basis, into an algorithm to find an orthogonal basis. Namely, given a symmetric matrix B of size $n \times n$, find a matrix $A$, such that $A^t B A$ is diagonal.
Lecture 17 (Oct 25): Sesquilinear form and Hermitian form.
video Lecture 18 (Oct 27): [LA] Sylvester theorem. Orthogonalization preserving flag.
video HW 10: 226, 229,230, 232, 233, 234, *239
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Lecture 19 (Nov 1): [LA] Cauchy-Schwarz. Adjoint map. Normal Operator.
video Lecture 20 (Nov 3): [LA] Proof of Spectral theorem for normal operator. Complexification of real vector spaces.
video HW 10: 253-257, 260, 261, 263,265
Lecture 21 (Nov 8) [ODE] 2.1.1 and 2.1.2
video No quiz on Nov 9 (wednesday)
Lecture 22 (Nov 10) [ODE] 2.1.2 and [LA] 4.2 Jordan form
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Lecture 24 (Nov 17) Constant Coeff Equation.
note ,
video HW: [ODE] (1) page 68, problem 2,4. (2) Page 115, ex 3.6.1, (3) page 119 ex 3.6.2
Lecture 25 (Nov 22) Boundary Conditions. Inhomogeneous Equations.
note,
video Midterm 2 Discussion (Nov 22 11:10-12:30).
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video HW: [ODE] p128, Ex 3.7.3
Thanksgiving
Lecture 26 (Nov 28) Fourier Series. Laplace equation and Heat equation.
note,
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