Lecture Outlines/Supplementary Material (Fall'10)

This webpage contains lecture outlines for Math115 (for the section taught by Prof. Henry Wolkowicz). Be aware that this webpage is continually being polished/changed.
(Also, see the course outline)

Lectures Date Subjects Covered Lecture Contents/Supplementary Materials
Lecture 36 Thurs. Dec. 2 Review
  • Review
    • Lecture 35 Wed. Dec. 1 Sect 6.2: Subspaces and spanning families; Sect 6.3: Linear Independence and Dimension
    • Lecture 34 Tues. Nov. 30 Sect. 6.1-6.2: Abstract vector spaces; subspaces
    Appendix A: Complex numbers;
  • Abstract vector spaces (definitions/examples); matrix spaces; function spaces
  • Complex numbers
  • Problems in text: Sect 6.1: 2,4,5
    App A: 1,2,3,4,5,6
    • Lecture 33 Thurs. Nov. 25 Sect. 8.2 Orthogonal diagonalization
  • symmetric matrices: have real eigenvalues; can be orthogonally diagonalized.
  • Examples: finding eigenvectors/eigenspaces by exploiting the orthogonality
  • Problems Sect 8.2: 1,4,5,11
    • Lecture 32 Wed. Nov. 24 Sect 8.1: orthogonal projections; Sect 8.2: orthogonal diagonalizations
  • orthogonal projection as a *symmetric* linear operator; orthogonal complements; decomposition of space Rn into U+U&perp
    • (make-up) Lecture 31 Tues. Nov. 23 Sect 8.1
  • orthogonality: orthogonal bases
  • specific homework problems: due Monday Nov. 29: Sect 8.1: 1c), 2c), 3a),3b); Sect 8.2: 1g), 5e),5f); Sect 6.1: 2c)
    • Lecture 30 Tues. Nov. 23 Sect. 5.5, 8.1
  • eigenvalue multiplicity, dimension of eigenspace, diagonalizability Problems in text: Sect 5.5: 1,5,8,9,10
  • Orthogonal Lemma
  • Problems in text: Sect 8.1: 1,2,3,5,9
    • Lecture 29 Thurs. Nov. 18 Sect. 5.5
  • Diagonalization revisited; equivalence of diagonalization with existence of a basis of eigenvectors
  • similarity definitions, examples, properties
  • not covered due to time constraint: diagonalizability and (multiplicity of eigenvalues - dimension of eigenspace) e.g. Theorem 6!
    • Lecture 28 Wed. Nov. 17 Sect 5.4-5.5
  • definition of rank A (dim row A/dim col A)
  • Rank Theorem (basis for row space from RREF; basis for col space from original A)
  • rank(AB) &le min(rank(A),rank(B))
  • basis for null(A) (using [I E] from RREF)
  • specific homework problems: sect 5.4 1a), 2c), 3c), 7a); sect 5.5 1e), 8c), 9c), 10a)c).
    • Lecture 27 Tues. Nov. 16 Sect. 5.4: Rank/RREF
  • elementary column operations do not change col(A), i.e. col(A) =col(AE) if E is invertible
  • elementary row operations do not change row(A), i.e. row(A) =row(EA) if E is invertible
  • Problems in text: Sect 5.4: 1a),2,3,7,10
    • Lecture 26 Thurs. Nov. 11 Sect. 5.3
  • Fourier expansion
    • Lecture 25 Wed. Nov. 10 Sect 5.3
  • problems: sect 5.3: 4,5a),6,10
    • Lecture 24 Tues. Nov. 9 Sect. 5.3: orthogonality
  • quick review of (including definitions): subspace; span; basis (including example of finding a basis for null A that ended up having orthogonal vectors)
  • Geometry: dot-product, length, unit vectors
  • Problems in text: Sect 5.3: 1,2
    • Lecture 23 Thurs. Nov. 4 Sect. 5.2
  • linear independence, basis, dimension
    • Lecture 22 Wed. Nov. 3 Sect 5.1,5.2
  • null space, image space, eigenspace
  • linear independence
  • problems: sect 5.2: 2a) b), 3,4,5a), 6,13,17a)
    • Lecture 21 Tues. Nov. 2 Makeup lecture;
    sect 5.1 cont.. and sect 5.2
  • examples of subspaces/spanning set; special problem presented in class;
    • Lecture 20 Tues. Nov. 2 Sect. 5.1: Subspaces and Spanning Sets
  • Definitions of subspaces and spanning sets
  • Problems in text: 1,2,9,16,22
    • Lecture 19 Thurs. Oct. 30 Sect. 3.3, 4.4
  • diagonalization and diagonalizability
  • projections/reflections/rotations (in R3)
    • Lecture 18 Wed. Oct. 28 Sect. 3.3 continued (eigs)
  • definition of eigenvalue/eigenvector: Ax= c x, x NOT 0!; characteristic polynomial;
  • skip: invariant subspaces; dynamical systems
  • specific problems: sect 3.3: 14,16,19,26
    • Lecture 17 Tues. Oct. 26 Sect. 3.3: Eigenvalues/Eigenvectors
  • Examples of eigenvalues/eigenvectors;
  • Problems in text: in Sect 3.3: 3,8,14,16,19,21,23,27 (specifically do 3,8!)
    • Lectures 14-15-16 Oct. 12,13,14 Sect. 3.1 determinants/interpolating polynomials 
  • Tuesday
 + determinants and elementary row operations
 + product rule ( det(AB) = det(A) det(B) )
 + transpose rule
 + inverse rule ( A invertible iff det(A) ~= 0 )

Wednesday
 + Adjugate
 + Adjugate formula for inverse
 + Cramer's rule

Thursday
 + Polynomial interpolation
 + definition of eigenvalues and eigenvectors
    • Lecture 13 Thurs. Oct. 7 Sect. 3.1 determinants
  • determinant properties
    • Lecture 12 Wed. Oct. 6 Sect. 2.6: Linear Transformations
    Sect. 3.1: Determinants
  • finding the matrix representations of linear transformations (using the standard basis); matrix representation of composite transformations; matrix representations of rotations
  • definition of determinant using cofactors/expansion along a row (Laplace expansion)
  • Problems in text: in Sect. 3.1: 2,4,5a),6,7,10,17
    • Lecture 11 Tues. Oct. 5 (supplementary) Sect. 2.6: Linear Transformations
  • Review of elementary matrices role in finding matrix inverses
  • properties linear transformations (black box)
  • Problems in text: in Sect 2.6: # 1,3,7,8a),9,17
    • Lecture 10 Tues. Oct. 5 Sect. 2.5: Elementary Matrices; Sect. 2.6: Linear Transformations
  • Review of block matrix multiplications
  • elementary matrices definition
  • Problems in text: in Sect 2.5: # 1,2,3a), 7a), 8b), 9
    • Lecture 9 Thurs. Sept. 30 Sect. 2.4: Matrix inverse
  • Problems in text in Sect 2.4: #1,2,3,4,9,16,22,28,30
    • Lecture 8 Wed. Sept. 29 Sect. 2.3 Matrix multiplication and block multiplication
  • Problems in text: Sect 2.3: 2,6,21,27,36 (and 11,12,13)
    • Lecture 7 Tues. Sept. 28 Sect. 2.1,2.2: Matrix Algebra
  • Matrix addition, multiplication, scalar multiplication
  • Problems in text: in Sect 2.1: # 2,3,6,12,13,20 in Sect 2.2: # 1,2,4a)b),6,7,11,12,13
    • Lecture 6 Thurs. Sept. 23 Sect. 1.3
  • examples for GE and GJE; rank; homogeneous systems
  • Problems in text in Sect 1.3: #1,2,7,9
    • Lecture 5 Wed. Sept. 22 Sect. 1.2
  • Gaussian (Gauss-Jordan) elimination; reduced echelon form (REF); row reduced echelon form (RREF);
  • Problems in text: Sect 1.2: 1,2,5, 9, 11, 12, 13, 16, 21
    • Lecture 4 Tues. Sept. 21 Sect. 4.2: Projection and Nearest Points
    Sect 1.1: Solving Linear Equations
  • Nearest points; projections; orthogonality
    • Given a point P and a line P0+ td, t in R: use orthogonality (projection) to find the closest point from P to the line. (first, find the projection of one vector on another vector).
    • examples
  • Solving Linear Equations
    • Definitions of: linear equation; coefficients; solutions; consistent system; inconsistent system; parametric form of general solution; augmented matrix.
    • start of: using elementary row operations for solving a linear system of equations
      1. interchange rows
      2. multiply a row by a nonzero scalar
      3. subtract a nonzero multiple of one row from another row
    • Problems in text: in Sect 4.2: # 12, 21, 25, 29, 44; in Sect 1.1: # 1,2;
    • Lecture 3 Thurs. Sept. 16 Sect. 4.1,4.2
  • Lines in Space
  • inner/dot product
  • angle between vectors (using the Law of Cosines)
  • Problems in text in Sect 4.2: #1,3,8,11
    • Lecture 2 Wed. Sept. 15 Sect. 4.1: Vectors and Lines (vector: addition and scalar multiplication)
  • Vector: addition and scalar multiplication
  • intrinsic descriptions: e.g. Theorem 1
  • Parallelogram Law: Theorems 2 and 3 and 4
  • Lines in Space: P+td (point P and direction d)
  • Problems in text in Sect 4.1: #1,4,7,14,21,22,24a), 24c)
    • Lecture 1 Tues. Sept. 14 Sect. 4.1: Vectors and Lines
  • Math. Prep. Information, i.e. for Mechanical EM15-Stream8, Tues. Sept 14, 7-8:15 PM
    A-L writing in RCH 110; M-Z writing in RCH 112
  • Alternate course webpage off of Henry Wolkowicz' webpage; includes important links and applications (e.g. to: search tools such as google; compression tools; etc...)
  • Course material covered:
    • (Geometric) Vectors in R2 and in R3; magnitude and direction; tip and tail of a geometric vector;
    • Theorem 1: four properties
    • vectors can be looked at in two ways: Geometric and Algebraic. E.g. In R3 it is a 3 by 1 matrix, i.e. a 3 by 1 array of numbers. (Note that matrices will be dealt with in detail as the course continues.)