Gaussian elimination, matrix operations, LDU factorization, inverses. Vector spaces, basis and dimension, the fundamental subspaces of a matrix. Linear transformations, matrix representations, change of bases. Orthogonality, Gram-Schmidt method, QR factorization, projections, least squares. Determinants, properties and applications. Eigenvalues and eigenvectors, diagonalization of a matrix, similarity transformations, symmetric matrices, applications to difference equations and differential equations. The Jordan form.
30% Hwk/Quizzes/Worksheets/Projects + 40% Exams + 30% FinalEach homework/quiz will carry the same weight.
Once you work out enough examples by hand and are comfortable with how things work, you may speed up some calculations by using software: Matlab, Mathematica, Maple, Sage (free) or even online tools like http://www.bluebit.gr/matrix-calculator, http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgiThere is also lots of videos explaining various topics from linear algebra, including the entire Introductory Linear Algebra course at MIT by G. Strang.
- 26-Aug/2019 | Linear systems | I || Vectors | II |
- Gaussian elimination One.I.1 Suggested exercises: pg 9: # 1.17, 1.19, 1.20, 1.22, 1.23, 1.24, 1.27, 1.31, 1.35
- Solution set One.I.2 Suggested exercises: pg 20: #2.20, 2.22, 2.23, 2.24, 2.25, , 2.27, 2.28, 2.29
- Solution set One.I.3 Suggested exercises: pg 33: #3.15 or 3.16, 3.18, 3.20, 3.21
- Vectors One.II.1 Suggested exercises: pg 41: #1.3, 1.4, 1.5, 1.7, 1.8,
- Vectors One.II.2 Suggested exercises: pg 47: #2.12, 2.14, 2.26, 2.27, 2.28, 2.31, 2.33,
- 2-Sep/2019 (no class Monday) | REF(A) | III |
- Gauss-Jordan elimination One.III.1 Suggested exercises: pg 54: #1.9, 1.10, 1.12
- Reduced echelon form One.III.2 Suggested exercises: pg 62: #2.11, 2.14, 2.24
- Using symbolic software [Hefferon Suggested exercises: pg 66: 1, 4.]
- Using numeric software [Hefferon Suggested exercises: pg 70: #3, 4, 5]
- Applications :
- Leontief model [Lay 1.6]
- Balancing Chemical equations [Lay 1.6]
- Balancing a diet [Lay 1.10]
- Networks and Kirkchhoff's laws [Hefferon pg71 Suggested exercises: pg 74: #1, #3, #4], [Lay 1.6+1.10, Williams 1.7, Strang 2.5]
- Vector spaces | I | Two.I.1 Suggested exercises: pg 86: #1.19, 1.22, 1.28
Additional Examples of vector spaces
- The set of solutions of homogeneous differential linear equations, like $e^x \frac{d^3y}{dx^3}+\cos x \frac{d^2y}{dx^2} + x \frac{dy}{dx}+ 127 y =0$, where $y=y(x)$ is an unknown function.
- The set of solutions of a linear homogeneous recursion, such as the Fibonacci recursion $F_{n+2}=F_{n+1}+F_{n}$, or a recursion $(n+2)(n+1)a_{n+2}=a_{n-1}$ that might arise from solving a differential equation by the power series method.
- 9-Sep-/2019
- Subspaces and spans Two.I.2 Suggested exercises: pg 97: #2.20, 2.21, 2.23, 2.24, 2.25, , 2.27, 2.29
- Linear independence | L | Two.II.1 Suggested exercises: pg 109: #1.20, 1.21, 1.24, 1.28, 1.35, 1.36, 1.42, 1.43
- Basis Two.III.1 Suggested exercises: pg 118: #1.18, 1.19, 1.20, 1.22, 1.23, 1.24, 1.25, 1.29, 1.31, 1.32, 1.33, 1.34, 1.37
- Dimension Two.III.2 Suggested exercises: pg 125: 2.16, 2.18, 2.21, 2.22, 2.29
- Vector spaces and Linear systems Two.III.3 Suggested exercises: pg 133: 3.17 - 3.21, 3.23, 3.32
- Isomorphism Three.I.1 Suggested exercises: pg 172: #1.13, 1.15, 1.16, 1.17, 1.18, 1.22, 1.23, 1.24, 1.34, 1.35
- Isomorphism Three.I.2 Suggested exercises: pg 181: #2.10, 2.11, 2.13, 2.14, 2.20, 2.25
- 16-Sep/2019
- Linear transformations |L|Three.II.1 Suggested exercises: pg 188: 1.18, 1.19, 1.20, 1.25, 1.30,
- Range and null space Three.II.2 Suggested exercises: pg 200: 2.21, 2.22, 2.23, 2.25, 2.26, 2.31, 2.35
- Matrix representation| Three.III.1 Suggested exercises: pg 211: 1.12, 1.16, 1.18, 1.20, 1.21, 1.26, 1.29, 1.33
- Matrix representation Three.III.2 Suggested exercises: pg 220: 2.14, 2.15, 2.16, 2.19
- Voting Paradoxes Suggested exercises: pg 155: #6
- 23-Sep/2019
- Matrix operations Three.IV.1 Suggested exercises: pg 226: 1.8, 1.13,
- Matrix multiplication Three.IV.2 Suggested exercises: pg 233: 2.14, 2.19, 2.21, 2.28, 2.34,
- Matrix multiplication Three.IV.3 Suggested exercises: pg 243: 3.28, 3.31, 3.34, 3.46
- Inverses Three.IV.4 Suggested exercises: pg 252: 4.13, 4.15, 4.28, 4.29, 4.34
- 30-Sep-19:
- Review
- Exam 1
- 7-Oct/2019 (no class Fri, Oct 11)
- Change of basis Three.V.1 Suggested exercises: pg 257: 1.7, 1.10, 1.17, 1.20
- Change of basis Three.V.2 Suggested exercises: pg 265: 2.10, 2.17
- Applications
- Colors, Cryptography [Williams 2.4]
- Markov chains [Hefferon Suggested exercises: pg 308: #5]
- Leontief model [ Williams 2.7, Lay 2.6, Leon 6.8, Strang 5.3]
- 14-Oct/2019 Notes with exercises!
- Orthogonal projection Three.VI.1 Suggested exercises: pg 270: 1.6, 1.17, 1.20
- Orthogonal projection Three.VI.3 Suggested exercises: pg 283: 3.10, 3.11, 3.14,
- 21-Oct/2019 Notes (continued)
- Applications:
- Digitization, and compression
- Lines of best fit [ Suggested exercises: pg 290: #7, 8, 9]. (Curves of best fit?)
- Inner products and Fourier series [Strang 3.4]
- Least square solutions [Lay 6.5]
- Orthogonal polynomials [Leon 5.6]
- Gram-Schmidt orthogonalization Three.VI.2 Suggested exercises: pg 277: 2.12,
- Determinants Four.I.2 Suggested exercises: pg 326: # 2.8, 2.9, 2.11, 2.15, 2.18 pg 321: # 1.1, 1.3, 1.4, 1.5, 1.7, 1.9, 1.16, 1.17, 1.19
- 28-Oct/2019
- Laplace expansion Four.III.1 Suggested exercises: pg 357: 1.14,
- Complex numbers Five.I.1, Five.I.2
- Similarity Five.II.1 Suggested exercises: pg 391: 1.7, 1.9, 1.10, 1.11, 1.20+1.21,
- Diagonalization Five.II.2 Suggested exercises: pg 396: 2.8, 2.11, 2.15,
- Eigenvalues and eigenvectors Five.II.3 Suggested exercises: pg 405: 3.23, 3.26, 3.28, 3.30, 3.32, 3.39, 3.40, 3.42
- 4-Nov/2019
- Review, Exam 2
- 11-Nov/2019( no class Monday)
- Applications of diagonalization Suggested exercises: Hefferon pg 3462: #2
- Difference equations and $A^n$ [Strang 5.3, Lay 5.6, Hefferon pg 458]
- System of difference equations of order 1 for $k$-dimensional vector: $\vec{x}_{n+1}=A \vec{x}_n$ with initial condition $\vec x_0$ has solution $\vec{x}_n=A^n \vec{x}_0$. If $A=P\Lambda P^{-1}$ is diagonalizable, then $\vec x_n=P \Lambda^n P^{-1}\vec{x}_0$. For small values of $k$ this results in explicit formulas for the solutions. See also Matrix difference equation on Wikipedia.
- Difference equation of order $k$ of the form $y_{n+k}=a_1 y_{n+k-1}+\dots + a_k y_{n-k}$ with initial condition that specifies values of $y_0,y_1,\dots y_{k-1}$ reduces to the previous case by setting $\vec{x}_n=\begin{bmatrix} y_{n+k-1} \\ y_{n+k-2} \\\vdots \\ y_n\end{bmatrix}$. In particular, the components of $\vec{x}_0$ are the initial values $y_0,y_1,\dots y_{k-1}$. See also Linear Difference Equation on Wikipedia.
For example, the second order recursion $y_{n+1}=a y_n+ b y_{n-1}$ with initial values $y_0,y_1$ can be solved as vector recursion $\vec {x}_{n+1}=A \vec{x}_n$ of order 1 with $$ \vec{x_n}=\begin{bmatrix} y_{n+1} \\ y_n\end{bmatrix} \mbox { and } A=\begin{bmatrix} a & b \\ 1 & 0\end{bmatrix} $$ To do so, shift the index to write the recursion as $y_{n+2}=a y_{n+1}+ b y_{n}$
Similarly, recursion of order 3 with $y_{n+1}=a y_n+ b y_{n-1}+c y_{n-1}$ with initial values $y_0,y_1, y_2$ can be solved as vector recursion $\vec {x}_{n+1}=A \vec{x}_n$ of order 1 with $$ \vec{x_n}=\begin{bmatrix} y_{n+2} \\ y_{n+1} \\ y_n\end{bmatrix} \mbox { and } A=\begin{bmatrix} a & b & c\\ 1 & 0 & 0 \\ 0 &1 &0\end{bmatrix} $$ To do so, shift the index to write the recursion as $y_{n+3}=a y_{n+2}+ b y_{n+1}+c y_n$- 18-Nov/2019
- Systems of Differential Equations and $e^{At}$ [Strang 5.4, Lay 5.7, Bronson Ch 7, Leon 6.2, Notes?]
- (Other topics were moved down to last week of classes)
- 25-Nov/2019 (no class Fri)
- Hamilton-Cayley Theorem and its applications
Theorem: If $p(\lambda)=\det (A-\lambda I)$ then $p(A)=0$. Hefferon pg 426
- Difference equations and $A^n$ [Lay 5.6, Strang 5.3]
- Systems of Differential Equations and $e^{At}$ [ Lay 5.7, Bronson Ch 7, Strang 5.4, Leon 6.2]
Inner products and finite element method [Strang 6.5]- Markov chains and Alphabet [ Williams 3.5, Lay 10.2, Leon 6.3, Hefferon pg 452]
Leontief model[ Williams 2.7, Lay 2.6, Leon 6.8, Strang 5.3]- Leslie matrices [Leon 1.4, Lay 4.8, Williams 3.5, Hefferon pg 456]
- 2-Dec/2019
- Orthogonal matrices Suggested exercises: Hefferon pg 316: #1
- Diagonalization of symmetric matrices
- Application: Quadratic forms [Williams 5.4, Lay 7.2, Notes]
Application: Singular value decomposition[Lay 7.4+7.5]Application: Buckling [Leon 6.2]- 9-Dec/2019: Final Exam Friday, December 13 10:30 a.m. - 12:30 p.m.
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