MATH 304-4 Fall 2019 SYLLABUS for Prof. Feingold's Section 4

Extra topics if time allows (usually not enough time for these in the elementary linear course) :

Topics which may be covered on each exam

These paragraphs will be updated before each exam is given.

Topics which may be covered on Exam 1

Here is a list of topics covered in lectures which may be covered on Exam 1.

Linear Systems, solving by row reduction of the augmented matrix [A|B] and interpretation in terms of free and dependent variables.

Consistent vs. inconsistent systems. Homogeneous systems AX=O.

Elementary row operations and reduction to Reduced Row Echelon Form (RREF).

Matrices, the set of all mxn real matrices, R^m_n, addition of matrices, multiplication of a matrix by a real number (scalar).

Matrix shapes, names of special patterns.

Rank of a matrix.

How an mxn matrix A determined a function L_A: Rⁿ --> R^m.

Properties of general functions: one-to-one (injective), onto (surjective), both (bijective), invertible.

Connection between properties of matrix A and function L_A.

Abstract definition of a real vector space, V. Examples, R^m_n is a vector space. For any set S, the set F = {f : S ---> R} of all functions from S to the reals R, is a vector space.

Definition of a linear transformation L : V ---> W from a vector space to a vector space. Ker(L), Range(L) = Im(L).

Basic facts about vector spaces and about linear transformations, and examples.

Defintion of matrix multiplication AB through the definition L_A composed with L_B equals L_AB. Lemma that L_A = L_B iff A=B.

Defintion of standard basis vectors e₁, ... , e_n in Rⁿ and lemma that Ae_j = Col_j(A), so AX is the sum of x_j Col_j(A).

Definition of positive powers of a square matrix A, and positive powers of L: V ---> V. Definition of transpose of a matrix.

Definition and some examples of subspaces.

Topics which may be covered on Exam 2

Topics that have been covered since Exam 1 and which may appear on Exam 2 are listed below.

Definition of when a square matrix is invertible, uniqueness of the inverse when it exists, and an algorithm to decide and find it by row reduction of [A | I_n].

Elementary matrices and how they can be used to achieve elementary row or column operations.

The rules of matrix algebra.

The span of a set of vectors S in a vector space V, and why it forms a subspace of V.

How to check that a subset W in V is a subspace of V.

Linear indepdendence or dependence of a subset of V, definition and method of determining that.

Theorems and examples about spanning and independence, connection with rank of a matrix.

Definition of a basis for a vector space, and how to decide if a subset is a basis of V.

Finding a basis for important examples of subspaces, Ker(L), Range(L), where L:V---> W is a linear map.

Use of a basis S of V to give coordinates with respect to S for each vector v in V. How that coordinate function, [v]_S, is a linear map from V to Rⁿ when a basis S for V consists of n vectors.

Transition matrices which give the relationship between the coordinates of a vector v with respect to different bases. If S and T are two bases of the same vector space, V, then the transition matrix from S to T is the square invertible matrix _TP_S such that [v]_T = _TP_S [v]_S.

Dimension of V as the number of vectors in any basis for V.

The standard basis for several examples of vector spaces, including all the R^m_n examples.

Row-space and Column-space of a matrix, and their dimension related to the rank of the matrix.

Information about the linear transformation L_A: Rⁿ--> R^m associated with rank(A).

The relationship between the dimensions of Ker(L), Range(L) and V for L:V---> W.

Extending an independent set to a basis, cutting down a spanning set to a basis.

How to represent a general linear map L:V---> W with respect to a choice of basis S in V and basis T in W by a matrix, that is,

using coordinates with respect to S, [ . ]_S, and coordinates with respect to T, [ . ]_T, to find a matrix _T[L]_S, such that _T[L]_S [v]_S = [L(v)]_T.

The algorithm for finding that matrix by a row reduction of [T | L(S)].

If S and S' are two bases of V, and T and T' are two bases of W, and L:V---> W then there is a relationship between _T[L]_S, the matrix representing L from S to T, and _T'[L]_S', the matrix representing L from S' to T'.

That relationship is _T'[L]_S' = _T'Q_{T T}[L]_{S S}P_S' where _SP_S' is the transition matrix from S' to S, and _T'Q_T is the transition matrix from T to T'.

The concept of isomorphism (bijective linear map) and its properties.

Material on determinants, their definition using permutations or by cofactor expansion, their properties, and methods of calculating them (definition by permutations or by cofactor expansions, crosshatching method for matrices of size n = 2 or n = 3 ONLY, using row operations).

The use of determinant to get the characteristic polynomial, det(A - tI_n), whose roots give the eigenvalues of A, and whose expression as a product of powers of distinct linear factors gives the algebraic multiplicities.

Eigenspaces, their properties, and how to decide if a matrix can be diagonalized or not. Theorems about eigenspaces and diagonalizability.

Independence of the union of bases for distinct eigenspaces.

Geometric multiplicity and its relationship to algebraic multiplicity for each eigenvalue.

Topics which may be covered on Exam 3

Eigenspaces and diagonalization may be tested again.

Section 6.1: The standard dot (inner) product in Rⁿ, length ||v|| and distance ||u-v||, orthogonality: u is perpendicular to v iff u.v = 0, S^⊥ and W^⊥, and the theorem that S^⊥ = (span(S))^⊥. cos(\theta_{u,v}) = (u.v)/(||u|| ||v||) formula for cosine of the angle between two vectors in Rⁿ using dot products.

Section 6.2: Orthogonal and orthonormal subsets, projection of a vector onto another vector, Proj_v(u) = (u.v)/(v.v) v. The geometrical meaning of projection may be helpful but I would not test it on an exam. Skip decomposing forces.

Section 6.3: The projection ṽ = Proj_W(v) of vector v into subspace W of Rⁿ defined to be a vector in W such that v - ṽ is orthogonal to W. If T = {w₁, ..., w_m} is any basis of W, then ṽ = ∑_i=1^m x_i w_i must satisfy (v - ṽ).w_j = 0 for j = 1,..., m, that is the linear system ∑_i=1^m x_i (w_i.w_j) = v.w_j for j = 1,..., m. Define the mxm symmetric matrix A = [w_i.w_j] and the mx1 column matrix B = [v.w_j]. This system is just AX = B, and it can be shown that A is invertible, so it always has unique solution X = A^-1 B which gives the projection. When basis T is orthogonal, the matrix A is diagonal and the solution is easy giving an explicit formula Proj_W(v) = ∑_i=1^m (v.w_i)/(w_i.w_i) w_i which is even simpler with T orthonormal. The Best Approximation Theorem, which says that ṽ = Proj_W(v) is the unique vector in W such that ||v - ṽ|| < ||v - w|| for any w in W distinct from ṽ. Skip Theorem 10.

Section 6.4: Gram-Schmidt process but skip QR factorization.

Section 6.5: Skip Least Squares completely.

Section 6.6: Skip Applications to Linear Models completely.

Section 6.7: Inner Product Space (IPS) and the Cauchy-Schwarz inequality, including the examples of function spaces C[a,b].

Section 6.8: Skip completely.

Topics which may be covered on the comprehensive Final Exam

Section 7.1: Diagonalization of symmetric matrices. Cover main ideas, examples, but the proof is beyond the scope of this course. The proof that eigenspaces of A = A^T from distinct eigenvalues are orthogonal is quite simple and accessible. Gram-Schmidt gives a way to get orthonormal basis for each e-space, so that diagonalization can be achieved by an orthogonal transition matrix P whose inverse is P^T. The Spectral Theorem is really advanced linear algebra, and the proof is beyond this course. I might mention it as a very nice result which only applies to symmetric real matrices. I would skip the Spectral Decomposition Theorem.

Skip Section 7.2 because some instructors did not have time to cover this material. For future reference, details about the topics from Section 7.2 were as follows: Quadratic forms are defined by Q_A(X) = X^T A X for X in Rⁿ where A = A^T. We may have time to cover basic properties and the effect of a change of basis X = PY for invertible P, Q_A(X) = (PY)^T A (PY) = Y^T (P^T A P) Y = Q_B(Y) where B = P^T A P is another symmetric matrix representing the same quadratic form. This begs the question: When can we find a nicer B? Can we find a diagonal B? The methods just completed about orthogonal diagonalization of a real symmetric matrix tell us the answer is yes.

Anything covered after the cutoff date for Exam 3 material.

Anything from any part of the course.

This page last modified on 12-17-2019.