Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 67: Modern Linear Algebra
https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf
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Lec # |
Topics |
Comments/Topics |
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1 |
What is linear algebra? |
Chapter 1 |
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2, 3 |
Complex numbers |
Chapter 2 |
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Discussion: Calculations with complex numbers; encoding linear systems |
Covers L1-3; 12.1 |
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4 |
Fundamental theorem of algebra (proof optional) |
Chapter 3 |
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5 |
Vector spaces and subspaces |
Chapter 4.1 - 4.3 |
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6 |
Direct sum, linear span |
Chapter 4.4 – 5.1 |
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Discussion: Vector space of matrices and operations on matrices |
Covers L5, 6; 12.2 |
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7 |
Linear independence of vectors |
Chapter 5.2 |
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8 |
Bases and dimensions of vector spaces |
Chapter 5.3 – 5.4 |
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9 |
Linear maps |
Chapter 6.1 |
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Discussion: Linear independence, homogenous linear systems, Gaussian elimination |
Covers L7, 8; 12.3.1-2 |
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10 |
Null space and range of linear maps |
Chapter 6.2 – 6.4 |
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11 |
Dimension formula for a linear map |
Chapter 6.5 |
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12 |
Matrix of a linear map |
Chapter 6.6 |
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Discussion: Linear maps, inhomogeneous systems, LU-factorization |
Covers L9-12; 12.3.3-4 |
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13 |
Invertibility |
Chapter 6.7 |
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14 |
Midterm |
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15 |
Eigenvalues and eigenvectors |
Chapter 7.1 – 7.3 |
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Discussion: Linear maps |
L12, 13; 12.4 |
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16 |
Existence of eigenvalues |
Chapter 7.4 |
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17 |
Upper triangular matrix representation |
Chapter 7.5 |
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18 |
Diagonalization (2x2) and applications |
Chapter 7.6 |
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Discussion: Eigenvalues and eigenvectors, special operations on matrices |
L15-18; 12.5 |
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19 |
Permutations and the determinant |
Chapter 8.1 – 8.5 |
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20 |
Properties of the determinant |
Chapter 8.6 – 8.7 |
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21 |
Inner product spaces |
Chapter 9.1 – 9.2 |
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Discussion: Calculation of the determinant, inner product spaces |
L19-21 |
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22 |
Cauchy-Schwarz, triangle inequality, Pythagoras |
Chapter 9.3 |
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23 |
Orthonormal bases, Gram-Schmidt procedure |
Chapter 9.4 – 9.5 |
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24 |
Orthogonal projections, minimization problems |
Chapter 9.6 |
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Discussion: Gram-Schmidt procedure and orthogonal projections |
L22-24 |
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25 |
Change of bases |
Chapter 10 |
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26 |
Self-adjoint and normal operators |
Chapter 11.1 – 11.2 |
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27 |
Spectral theorem for normal maps (complex) |
Chapter 11.3 |
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Discussion: Change of basis, diagonalization |
L25-27 |
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28 |
Diagonalization |
Chapter 11.4 |
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29 |
Positive operators, polar and singular value decompositions |
Chapter 11.6 – 11.7 |
https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf