線性代數

線性系統、矩陣與向量分析、特徵值與特徵向量

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4.教學方法及教學相關配合事項(如網站、助教、圖書講義及資料庫等):e3 system

1學期作業:無 2.考試狀況:Quiz #6 (60%), med-term (20%), final (20%)

• Introduction to Linear Algebra (1) The Geometry of Linear Equations (2.1)

  row picture column picture matrix form in linear equation

• Elimination (2.2) Matrix Operation Elimination using Matrices (2.3)

  idea of elimination; back substitution row/column operation row reduction for elimination

• Rule of Matrix Multiplication (2.4) Inverse Matrices (2.5)

  Matrix-vector operation 4 ways to see the matrix multiplication inverse Gauss-Jordan Elimination

• Factorization A = LU (2.6) Transposes and Permutations (2.7)

  production of elimination matrices matrix factorization (LU) RTR and RRT; PT=P-1

• Vector Spaces (3.1)

  Rn space; vector, linear combination, space, subspace, matrix form C(A)

• Column Space C(A) and Null Space N(A) (3.2)

  Ax = b and Ax = 0

• Rank and the RREF (3.3)

  computing nullspace (Ax = 0) pivot number (rank); free variables special solutions rref(A) = R

• Complete Solution (3.4)

  Ax= b; x= xp+xn; Matrix analyses (4 full rank cases)

• Independence, Basis and Dimension (3.5)

  Linear independence; spanning a space; basis for a vector space; dimension

• 4 fundamental subspaces (3.6)

• Bases of new vector spaces Rank one Matrices Small World Graphs

• Applications

  Graphs and Networks Incidence matrix KCL

• Orthogonal vectors and subspace (4.1)

  Orthogonal vectors Subspace Row space orthogonal to nullspace Why projection Ax = b no solution

• Projection (4.2)

  Projection Projection matrix in 1D Why projection Ax = b no solution in 2D

• Least square approximation (4.3)

  Projection matrix Great picture Least square approximation ATA invertible (4G)

• Orthogonal Bases and Gram-Schmidt (4.4)

  Orthogonal bases Orthogonal matrix Q Gram-Schmidt A Q Factorization A = QR

• Determinants (5.1)

  10 properties

• Permutations and cofactors (5.2)

  Formula for detA Big formula Cofactors formula Special Tridiagonal matrices

• Cramer’s rule, Inversion and Volumes (5.3)

• Eigenvalues and Eigenvectors (6.1)

  det[A-I] = 0 Trace = 1+2+... Determinant = 12...

• Diagonalizing a Matrix (6.2)

  A = SS-1 Power of A Fibonacci numbers

• Differential Equations (6.3)

  eAt of a matrix

• Markov Matrices (8.3) and Fourier Series (8.5)

  Markov Matrices (probability) Fourier Series (orthogonal projection)

• Symmetric matrices (6.4)

  A = QQT

• Complex vectors of Matrices (10)

  Complex inner product Hermitian and Unitary

• Positive Definite Matrices (6.5)

  Tests of positive matrix Minimum xTAx > 0 Ellipse equation ATA

• Similar Matrices (6.6)

  B = M-1AM

• Singular Value Decomposition (SVD) (6.7)

  A = UVT

Gilbert Strang, Introduction to Linear Algebra.

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