MATH 304-3 Fall 2021 SYLLABUS for Prof. Feingold's Section 3

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

The list of topics below is most closely related to my lecture notes, but some students have asked to know which sections of the textbook (Heffron) contain these topics, so I am listing those here: Ch. 1, I. 1-3 (linear systems), III. 1-2 (RREF), Ch. 2, I. 1-2 (vector spaces), II. 1 (linear independence), III. 1-3 (basis, dimension), Ch. 3, I. 1-2 (linear maps), II. 1-2, III. 1-2 (matrix representation of maps), IV. 1-4 (Matrix algebra).

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).

The span of a set of vectors in R^m_n as the set of all linear combinations from that set.

Matrix shapes, names of special patterns.

Rank of a matrix.

How an mxn matrix A determines a function L_A: Rⁿ --> R^m by L_A(X) = AX.

Linearity properties of the function L_A, that is, L_A(X+Y) = L_A(X) + L_A(Y) for any X, Y in Rⁿ, and L_A(rX) = r L_A(X) for any X in Rⁿ and any r in R.

Definition of Ker(L_A) and of Range(L_A) = Im(L_A) and how to find them by row reduction methods.

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

Connection between properties of matrix A and function L_A.

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.

Formula for the matrix product of an mxn matrix A with an nxp matrix B giving an mxp matrix C = AB whose columns are A(Col_k(B)) for k = 1, ..., p.

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).

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 (maps), and examples.

Definition and some examples of subspaces.

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].

Definition of transpose of a matrix, of symmetric and anti-symmetric matrices.

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.

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 and the vector space of polynomials with degree at most k.

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).

Topics which may be covered on Exam 2

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.

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.

Topics which may be covered on Exam 3

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(tI_n - A), 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.

Definition and properties of the standard dot product in Rⁿ: bilinear, symmetric, positive definite.

Definition of length of a vector, ||v||, for v in Rⁿ.

Definition of distance between two vectors, ||u-v||.

Definition of the angle a between two vectors u and v in Rⁿ given by the formula cos(a) = u.v/(||u||)(||v||).

Definition of two vectors in Rⁿ being orthogonal (perpendicular) when the angle between them is a right angle (90 degrees = pi/2 radians), so u.v = 0.

The Cauchy-Schwarz inequality |u.v| <= (||u||)(||v||) and the triangle inequality ||u+v|| <= ||u||+||v||.

Orthogonal and orthonormal subsets, projection of a vector onto another vector, Proj_v(u) = (u.v)/(v.v) v.

A real nxn matrix A is called orthogonal when A^T = A^-1, its transpose is its inverse.

A real nxn matrix A is orthogonal iff the set of its columns is an orthonormal set in Rⁿ.

When A is orthogonal, the linear map L_A(X) = AX preserves lengths and angles because (AX).Y = X.(A^TY) for any X and Y in Rⁿ.

The geometrical meaning of projection may be helpful but I would not test it on an exam. Skip decomposing forces.

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.

The Gram-Schmidt (G-S) orthogonalization process for converting a basis T = {w₁, ..., w_m} of a subspace W to an orthogonal basis T' = {w'₁, ..., w'_m}.

The formula for each vector in T' is w'_j = w_j - Proj_W(j-1)(w_j) for j = 2, ..., m, where W(j-1) is the subspace with orthogonal basis {w'₁, ..., w'_j-1} and w'₁ = w₁ to start the process.

So from the explicit formula for such a projection, we have Proj_W(j-1)(w_j) = ∑_i=1^j-1 (w_j.w'_i)/(w'_i.w'_i) w'_i

The Pythagorean Theorem in Rⁿ which says that for any orthogonal set T = {w₁, ..., w_m}, we have ||w₁||² + ... + ||w_m||² = ||w₁ + ... + w_m||².

The standard dot product in the complex vector space Cⁿ which is sesquilinear, conjugate symmetric and positive definite.

If Z and W are column vectors in Cⁿ then Z.W = Z^T W^- where Z^T means the transpose of Z and W^- means the complex conjugate of W.

The Hermitian conjugate of a complex nxn matrix A is Ā^T, the transpose conjugate of A denoted by A^H.

For any column vectors Z and W in Cⁿ and any complex nxn matrix A, we have (AZ).W = Z.(A^HW).

A complex nxn matrix A is called Hermitian when A^H = A. It is called unitary when A^H = A^-1.

A is unitary iff the set of its columns is an orthonormal set in Cⁿ.

We proved that any real symmetric matrix A = A^T in Rⁿ_n has all its eigenvalues in the real numbers, and that it is diagonalizable by an orthogonal transition matrix P.

Topics which may be covered on the comprehensive Final Exam

We defined the perp of a subset S in Rⁿ to be S^⊥ = {X in Rⁿ | X.s = 0 for all s in S}.

We proved that S^⊥ = <S>^⊥, the perp of S equals the perp of the span of S. So to get the perp of a subspace, W, just find the perp of a basis for W.

To find the perp of a subset S = {v₁,...,v_m} in Rⁿ, just solve AX = 0 where A is the mxn matrix with Row_i(A) = v_i^T for i = 1,...,m.

We defined the sum of subspaces W = W₁ + ... + W_m in any vector space V.

We showed that sum W is a subspace of V, and defined it to be a direct sum when each vector w in W has a unique expression as w = w₁ + ... + w_m for w_i in W_i.

We discussed basic examples of sums and direct sums in R² and R³.

We discussed the theorem which says that a sum W (as above) is direct iff the intersection of W_i with the sum of all other W_j is only the zero vector, for each i = 1,...,m.

We discussed the theorem that if B_i is a basis of W_i, then the sum W is direct iff the long list B of all the vectors in B₁ ... B_m is a basis of W.

We defined a direct sum of subspaces in Rⁿ to be an orthogonal direct sum when those subspaces are mutually orthogonal, that is, W_i⊥W_j for i≠j.

We proved the theorem that for any subspace W in Rⁿ, that W + W^⊥ = Rⁿ.

We applied these ideas to the eigenspaces of a real symmetric nxn matrix A = A^T, and showed that Rⁿ is the orthogonal direct sum of all those eigenspaces.

We then got that any real symmetric nxn matrix A is orthogonally diagonalizable, D = P^TAP, for an orthogonal transition matrix P.

Anything covered after the cutoff date for Exam 3 material.

Anything from any part of the course.

This page last modified on 12-31-2021.