MATH 3000 (Azoff)  Fall 2009

Section 3.6:  Abstract Vector Spaces (up through Example 9, i.e., not responsible for inner products)

• Definition and subspaces
• Examples: spaces of matrices and spaces of functions
• Definitions of spanning, independence, and bases
• Testing independence in examples
• Finding bases in examples
  

Section 4.1: Inconsistent Systems and Projection onto a subspace V  of  Rn

• Proj_V b is the unique vector p in V satisfying (b-p) perp V.  Of all vectors in V, it is the one which is closest to b.
• review the definition of subspace V of Rⁿ and its orthogonal complement V^perp
• recall that V+V^perp = Rⁿ, and V intersect V^perp = {0}
• relation to matrices: R(A) and N(A) are orthogonal complements of each other 

Methods for computing  ProjV b:

• as in Chapter 1 when V is one-dimensional

• via a matrix A whose column space is V, that is, ProjV b= Ax where x satisfies the normal equation ATAx=ATb

• simplifying the above by using an orthogonal basis for V

• by formula when the columns of A are independent, that is,   Proj_V b= A(A^TA)^-1(A^Tb);
• by computing projection onto V^perp first

Using projections to compute reflections

Least squares and Applications to data fitting

Section 4.2: Gram-Schmidt Process (only thru Example 4.2, that is, not including the discussion of the QR decomposition on pages 201-202)

• Definition of orthogonal, orthonormal bases
• Constructing them
• Their advantages in computing coordinates, projections

• Review of definition and standard matrix [T]std

• Matrix of a linear transformation relative to a basis [T]beta 
  

• Change of basis formula

• Using matrices relative to appropriate bases to compute formulas and standard matrices for projections and reflections
  

• Definition and properties of similarity
  

• Examples of linear transformations on spaces of functions

• Kernels & images

Chapter 5 : Determinants

• Definition by characterizing properties [effect of row operations]

• Other properties 

               det (AB)=(det A)(det B) 

               det (AT)=det A 

               det A=0 if and only if A is singular

• Computation by row operations

• Shortcut for upper/lower triangular matrices

• Computation via cofactors
  

Section  6.1: Eigenvalues and Eigenvectors

• Definition of eigenvalues, eigenvectors & eigenspaces

• Characteristic polynomials and finding eigenvalues

• Finding a basis for E(t) = null[A-tI] 

Section  6.2: Diagonalizability

• Definition and the equations AP = PD ;  A= PDP^-1
• Using eigenvalues and eigenvectors to construct D and P
  
• Eigenvectors associated with distinct eigenvalues are independent
• Criteria for diagonalizability

Section  6.3: Applications

• Computing powers of diagonalizable matrices
• Class attendance example and related Markov processes