a s 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula

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1 Syllabus for Math 308, Paul Smith Book: Kolman-Hill Chapter 1. Linear Equations and Matrices 1.1 Systems of Linear Equations Definition of a linear equation and a solution to a linear equations. Meaning of the coefficient of x i. Definition of an m n system of linear equations (m linear equations in n unknowns) and a solution to a system of linear equations. Definitions of consistent and inconsistent systems, homogeneous systems, trivial and non-trivial solutions. Manipulating a system of linear equations. Method of elimination. Important summary of those methods on page 6. Definition: two systems of m linear equations in n unknowns are equivalent if they have the same solutions. The trichotomy: no solution; a unique solution; infinitely many solutions. The idea behind this trichotomy: if p and q are different solutions to the same system all the points on the line through p and q, which we denote pq, are solutions to that system: λp + (1 λ)q, λ R. Symbols N Z Q R C. Set notation as in the notes posted on my 308 web page. 1.2 Matrices Definition, m n matrix has m rows and n columns. Standard labeling for the entries: a ij is in row i and column j. Definition of equality of matrices. Square matrices, their diagonals. Column and row vectors. Definition of R n as all n 1 vectors/matrices. Or, sometimes simpler to define R n as all 1 n row vectors. Similarly for C n, row and column vectors whose entries are complex numbers. Addition of matrices (same size), associative, commutative. Multiplication of a matrix by a number, or scalar, c R, c(a ij ) = (ca ij ). Define A = ( 1)A. Meaning of A B. Definition of linear combination of columns. Coefficients. Summation notation n n n m n n m a i = a j = and a ij = a ij. i=1 j=1 s=1 a s i=1 j=1 j=1 i=1 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula n A ij = A ik B kj. k=1 Use this to prove the associative law (AB)C = A(BC). And to prove (AB) T = B T A T. Let A be an m n matrix and B a p q-matrix. Can form the product AB if and only if n = p. Know AB need not equal BA; have a few such examples at your fingertips. Can have AB = 0 and BA 0; have a few 1

2 2 such examples at your fingertips. In particular, the product of non-zero matrices can be zero. Know the very important formula Ax = x 1 A x n A n (1) where A i is the i th column of A. Know how to go back and forth between a system of linear equations and a single matrix equation Ax = b and how to form of the augmented matrix (A b) and its meaning. Know why (1) implies Ax = b if and only if b is a linear combination of the columns of A. 1.4 Algebraic Properties of Matrix Operations. You must know everything in this section. Relation between the dot product and matrix multiplication. 1.5 Special Types of Matrices and Partitioned Matrices. You must know the meaning of A r for all integers r. Meaning of diagonal matrix, the different identity matrices, and zero matrices, the meaning of symmetric and skew-symmetric matrices. The only partitioned matrix we care about is the augmented matrix. Definition of the inverse of a matrix (if it exists) and how to prove it is unique if it exists. Relation among inverses of various matrices, e.g., A 1 B 1 = (BA) 1 ; (A T ) 1 = (A 1 ) T. You should figure how to prove these proofs in book if needed. If A is invertible the unique solution to Ax = b is A 1 b. Why? On page 46 the book gives a different definition of a (non- )singular matrix than the one we are using in class. Use the definition I gave: a square matrix A is non-singular if and only if the only solution to Ax = 0 is x = 0. Because we use a different definition we must prove that a matrix is non-singular if and only if it has an inverse. The book doesn t need to do that because that is how they define non-singularity! 1.6 Matrix Transformations. VERY IMPORTANT to know that an m n matrix gives a linear transformation f : R n R m by the formula f(x) = Ax. Meaning of the words image and range. Special matrix transformations rotations about the origin in the plane; reflection with respect to the x- and y-axes; projection R 3 R 2 ; contraction and dilation 1.7 Computer Graphics. We won t cover this in class and it won t be on the exam but read it as part of your general education about matrices. 1.8 Correlation coefficients. We won t cover this topic but we will use some ideas introduced in this section. You need to know the formula for the length of a vector, and the cosine of the angle between two non-zero vectors; the criterion for two vectors to be perpendicular (or orthogonal): if u and v are non-zero vectors u is orthogonal to v if u v = 0;

3 3 1.? Odds and ends. You must know the meaning of a statement like P if and only if Q: two statements, P and Q, are involved and the proof will have two separate parts, one showing if P is true, then Q is true, the other showing if Q is true, then P is true. You must know that the following statements are equivalent: P implies Q; if P, then Q; if Q is false so is P ; if P is true so is Q. Chapter 2. Solving Linear Systems. 2.1 Echelon Form of a Matrix Definition of row echelon form and row reduced echelon form (rref). You must be able to state these cleanly and with 100% accuracy. Even the slightest inaccuracy will be penalized see answers to quiz 2. We will not make use of column echelon form and column reduced echelon form but you should know of their existence and that they are completely analogous to row echelon form and row reduced echelon form. You must know the 3 elementary row operations (EROs) and be able to state them cleanly and with 100% accuracy. If you can get from A to B by a sequence of EROs you can get from B to A by a sequence of EROs. What it means for two m n matrices to be row equivalent the book delays that definition until section 2.4. (There is a notion of column equivalent but we will not make use of it.) Theorem: Two matrices are row equivalent if and only if each can be obtained from the other by an ERO. Theorem: Every matrix is row equivalent to one in echelon form, and to a unique one in row reduced echelon form. We will not use the words pivot and pivot column 2.2 Solving Linear Systems Understand how to write down the solutions to Ax = b when (A b) is in row echelon form Suppose (A b) is equivalent to (E c) where E is in RREF. How to recognize from (E c) whether Ax = b is inconsistent. Definition of dependent and independent variables: x j is dependent if column j of E contains a leading 1. All other variables are independent. Independent variables can take on any value; once the independent variables are given particular values the values of the dependent variables are completely determined. Know how to write down all solutions to Ax = b by using (E c). 2.3 Elementary Matrices: Finding A 1 My treatment of inverses differs from that in the book. I do not make use of elementary matrices. The idea behind elementary matrices is quite simple. There are three types of elementary matrices. The book calls them Type I, Type II, Type III. Multiplying a matrix A on the left by one

4 4 of these elementary matrices, E say, produces a matrix EA that can also be obtained from A by an elementary row operation, and vice versa; if B is obtained from A by performing a single ERO there is an elementary matrix E such that EA = B. For example, consider the ERO swap row i with row j ; perform that operation on the m m identity matrix to produce the matrix E; then E is an elementary matrix; the matrix EA can be obtained by swapping rows i and j of A. Similarly for the other two EROs. Thus the result of performing a sequence of EROs on A produces a matrix B that is equal to E n E n 1 E 1 A where each E i is an elementary matrix (see Thm. 2.6). Using elementary matrices the book proves some of the same results we proved about invertible matrices. Lemma 2.1in the book is important: a matrix is non-singular if and only if it is row equivalent to the identity matrix.that is the content of Lemma 2.1 in simpler terms. They state that result as Corollary 2.2, but the proof of my version of Lemma 2.1 is simpler than their s. The blue box on page 120 is important. We have proved that in class, with the exception of (5). It is important to know how to find A 1 when it exists. Notice that the book uses exactly the same method as the one I showed you in class see their Example 4 on page 121. We also proved Theorem 2.11 as part of our proof that a nonsingular matrix is invertible. 2.4 Equivalent Matrices. I have already discussed some of this material in section 2.1 above. We have not and will not prove Theorem 2.12 but you will learn something useful by reading the proof carefully. Theorem 2.14 is important. It can be proved without using elementary matrices. (To do that combine Prop. 5.4, Thm. 5.5, and Thm. 9.7 in my notes.) 2.5 LU-Factorization (Optional) We will skip this section. Chapter 3. Determinants 3.1 Definition In class we define the determinant in a different way from the book. Consequently most of section 3.1 can be skipped (or postponed). But do read from Example 6 to the end of the section. And some of the exercises are worth doing. 3.2 Properties of Determinants Know the statements of all Theorems in this section, though you can skip the proofs. Read the examples. 3.3 Cofactor Expansion Read section 3.3 carefully. We use the cofactor expansion to define the determinant inductively, i.e., first we define it for a 2 2 matrix; then we assume we have a formula for the determinant of an (n 1) (n 1) matrix and define the determinant of an n n matrix in terms of the formula for an (n 1) (n 1) matrix. The precise formula

5 we use is given in the first part of Theorem 3.10 with i = 1 and a slight change to the meaning of A ij. For information about the sign see Defn Compare the formula at the bottom of page 158 with the formula we gave in class for a 3 3 matrix. Now look at Example 2. The material about computing areas will not be something I expect you to know, but it would be good to read through it as part of your education. 3.4 Inverse of a Matrix We will cover this section in detail. We have already seen one way to compute the inverse: form the augmented matrix (A I) then perform EROs to get this in the form (I B) (if A does not have an inverse this will not be possible). The matrix B is A 1. This section gives another way to compute the inverse of A by computing the determinants of its minors (see Defn. 3.3, p. 157, for the defn. of minor. 3.5 Other Applications of Determinants The idea in this section is simple: if A has an inverse, the equation Ax = b always has a unique solution, namely x = A 1 b. This is about as obvious as saying that the equation 3x = 7 has a unique solution, namely x = That is all there is to Cramer s Rule. The point is that even if we know A has an inverse we still have to compute it if we want to write down an explicit solution. Cramer s Rule just does that using the formula for A 1 given in sect Determinants from a Computational Point of View Chapter 4. Real Vector Spaces. 4.1 Vectors in the Plane and in 3-space 4.2 Vector Spaces 4.3 Subspaces 4.4 Span 4.5 Linear Independence 4.6 Basis and Dimension 4.7 Homogeneous Systems 4.8 Coordinates and Isomorphisms 4.9 Rank of a Matrix Chapter 5. Inner Product Spaces. 5.1 Standard Inner Product on R2 and R3 5.2 Cross Product in R3 (Optional) 5.3 Inner Product Spaces. 5.4 Gram-Schmidt Process. Orthogonal Complements. 5.5 Least Squares (Optional). Chapter 6. Linear Transformations and Matrices. 6.1 Definition and Examples. 6.2 Kernel and Range of a Linear Transformation. 6.3 Matrix of a Linear Transformation. 5

6 6 6.4 Vector Space of Matrices and Vector Space of Linear Transformations (Optional). 6.5 Similarity. 6.6 Inroduction to Homogeneous Coordinates (Optional). Chapter 7. Eigenvalues and Eigenvectors. 7.1 Eigenvalues and Eigenvectors. 7.2 Diagonalization and Similar Matrices. 7.3 Diagonalization of symmetric matrices Appendices. You should also know the material in the appendices.

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