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Systems of linear equations; matrix algebra, inverses, determinants, eigenvalues and eigenvectors, diagonalization; vector spaces, basis, dimension, rank and nullity; inner product spaces, orthonormal bases; quadratic forms.
This course has a two-fold aim:
(1) develop proficiency in the concepts listed above, and
(2) transition students into abstract mathematics thorugh development of good habits of understanding, communicating, and writing proof-based mathematics.
Required for AM majors, elective for other majors.
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MATH 251, multivariable calculus. In particular, some awareness of algebraic and geometric properties of vectors in 2- and 3-space.
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| Textbook
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Elementary Linear Algebra by Howard Anton, 10th ed., John Wiley and Sons
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- Students will learn how to solve systems of equations through various techniques,
especially through row reduction of matrices.
- Students will learn properties of matrices, especially invertibility, and matrix
algebra.
- Students will learn general vector spaces and linear transformations. The primary
vector spaces studied are Euclidean, matrix and polynomial spaces. Topics
include linear independence, column and row spaces, nullspace, basis and
dimension (including rank and nullity).
- Students will explore eigenvectors and eigenvalues and learn how to diagonalize a
matrix.
- Students will learn about inner product spaces and orthogonailty. Topics include
orthonormal bases, the Gram-Schmidt process, best approximation, and their
applications to QR-decomposition and least squares-fitting.
- Students will learn about applications of linear systems in sciences and
engineering.
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- Linear Systems and Matrices:
Elementary row operations, Gaussian elimination, elementary matrices and LU decomposition, matrix inverse and invertibility, determinant, applications in electrical circuits or chemical reactions.
- Euclidean Vector Spaces:
Vector algebra in n-dimensional real space R^n, dot product and orthogonality, linear transformations and geometric operators.
- General Vector Spaces:
Examples, including R^n, P_n, M_mn, and non-examples, subspaces and spanning sets, linear independence, basis and dimension, null, row and column spaces of a matrix, application to Markov chains.
- Eigenvalues and Eigenvectors:
Eigenvalues, eigenvectors, eigenspaces, and diagonalization,
application to Markov chains (contd.)
- Inner Product Spaces:
Inner products – examples, non-examples, and properties, orthonormal basis,
Gram-Schmidt process and application to QR-decomposition, best
Approximation and least squares problem, application to least squares fitting.
- Orthogonal Matrices and Quadratic Forms:
Orthogonal matrices. Orthogonal decomposition, quadratic forms and
positive/negative definite matrices, application to optimization.
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Reasonable accommodations will be made for students with documented disabilities. In order to receive
accommodations, students must obtain a letter of accommodation from the Center for Disability Resources
and make an appointment to speak with me [the instructor] as soon as possible. The Center for Disability
Resources (CDR) is located in Life Sciences Room 218, telephone 312-567-5744 or disabilities@iit.edu.
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