Math 107: Introduction to Linear Algebra, Springer 2018

Topics Covered in Lectures by Ali Mostafazadeh

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 06	Review of the content of the syllabus; some historical remarks on linear algebra, systems of linear equations, geometric demonstration of the existence and uniqueness of the solution of a system of two linear equations, coefficient and augmented matrix, examples of row operations for a system of two linear equations and their application for solving the system when there is a unique solution	Pages 18-28 of Lay et al, 5^th Edition
2	Feb. 08	Application of row operations for a system of two linear equations with infinitely many or no solutions, the method of Gaussian elimination applied for solving a system of three linear equations, realizing Gaussian elimination using row operations applied to the augmented matrix of the system, echelon and reduced echelon form of a matrix, general algorithm for bring a matrix to its echelon and reduced echelon form, application for deciding when a system has infinitely many solutions (free variables) and no solutions	Pages 28-40 of Lay et al, 5^th Edition
3	Feb.13	Addition and scalar multiplication in R^2 and their basic properties, proof of the distribution law for scalar multiplication over addition in R^2, linear combination and span of subsets of R^2, generalization to R^3 and R^n, the link between the characterization of the span of m vectors in R^n and the solution of a linear system of n equations in m unknowns	Pages 40-51 of Lay et al, 5^th Edition
4	Feb. 15	Multiplication of a matrix by a column and its basic properties, the matrix form of a system of linear equations, a criterion for the existence of a solution of the matrix form of a system of linear equations, homogeneous systems, their trivial and nontrivial solutions, the solution of non-homogeneous systems	Pages 51-66 of Lay et al, 5^th Edition
5	Feb. 20	Subspaces, linearly-dependent, and linearly-independent subsets of R^n	Pages 164-165 & 72-79 of Lay et al, 5^th Edition
6	Feb. 22	Linearly-independent subsets with one or two elements, bases of R^n, linear transformations, linear transformations from R to R, the zero and identity transformations, projections to a component	Pages 166-167 & 79-82 of Lay et al, 5^th Edition
7	Feb. 27	Reflections and rotations in R^2 as linear transformations, range and onto linear transformations: T:R^n->R^m; Theorem: T(0)=0; Theorem: Ran(T) is a subspace; 1-to-1 linear transformations; Theorem: T is 1-to-1 if and only if T(x)=0 implies x=0.	Pages 82-87 of Lay et al, 5^th Edition
8	Mar. 01	Standard matrix (representation) for a linear transformation T:R^n->R^m, examples; Definition of the equality of functions f:R^n->R^m, the addition of two such functions and multiplication of a real numbers by such functions; the properties of the addition and scalar multiplication of functions f:R^n->R^m; Theorem: Sum and scalar multiples of linear transformations T:R^n->R^m are linear transformations; Defining sum and scalar multiples of matrices of the same size as the standard matrix for the sum and scalar multiples of linear transformations; Properties of the addition and scalar multiplication of matrices	Pages 110-112 of Lay et al, 5^th Edition
9	Mar. 06	Defining the multiplication of a matrix T by a column vector x as the action of its linear transformation T, i.e., T x:=T(x); formula of the entries of T x and use of the summation symbol; Composition of linear transformation T:R^n->R^m and S:R^m->R^l, Theorem: Composition of two linear transformations is a linear transformation; Defining the product of two matrices S and T as the standard matrix for the composition of their linear transformations (SoT); derivation of the formula for entries of ST; list of properties of the matrix multiplications	Pages 112-115 of Lay et al, 5^th Edition
10	Mar. 08	Proof of some of properties of matrix multiplication, positive integer powers of a square matrix, the transpose of a matrix and the properties of transposition with proofs; inverse of a square matrix and invertible matrices; characterization of invertibility of a matrix A in terms of the existence of a unique solution for the matrix equation Ax=b; properties of inversion of matrices with proofs.	Pages 115-124 of Lay et al, 5^th Edition
11	Mar. 13	Assigning linear transformations to row operations, the (elementary) matrices associated with the basic row operations, an algorithm for computing the inverse of an invertible matrix; the linear transformation defined by the inverse of an invertible matrix	Pages 124-129 of Lay et al, 5^th Edition
12	Mar. 15	Various characterizations of invertibility of a square matrix and the corresponding linear transformation; Review of the concepts of subspace, range of a linear operator (column space of its standard matrix), and the connection to solutions of linear system of equations, null space of a linear operator, construction of a basis for the range and null space of a linear operator, Theorem: Let A: R^n -> R^m be a linear transformation, then dim(Null(A))+dim(Ran(A))=n (with a proof).	Pages 129-132 & 164-171 of Lay et al, 5^th Edition
Midterm Exam 2	Mar. 17
13	Mar. 20	Review of the definition of the dimension of a subspace of R^n; basic properties of the bases of a subspace U of R^n; the coordinates and coordinate vector of elements of U relative to a basis of U; rank and nullity of a matrix and the rank-nullity theorem, application to invertible matrices	Pages 171-179 of Lay et al, 5^th Edition
14	Mar. 22	Invertibility of a 2x2 matrix and its determinant; Determinant of a 2x2 matrix as an antisymmetric multilinear function of columns of the matrix; extension of the basic algebraic properties of the determinant of 2x2 matrices to nxn matrices; the Levi Civita symbol and the determinant of 3x3 matrices; extension to nxn matrices	Pages 181-184 of Lay et al, 5^th Edition & Pages 195-210 of Lang, 2^nd Edition
15	Mar. 27	Cofactor expansion of the determinant along rows; Theorems with proofs: 1) Swapping columns changes the determinant of the matrix by a sign; 2) Matrices with two identical columns have zero determinant; 3) Adding a multiple of a column to another column does not change the determinant of the matrix; 4) det A^T=det A; 5) Statements 1-3 are true if we change “column(s)” to “row(s).”	Pages 184-188 & 190 of Lay et al
16	Mar. 29	Review of basic properties of determinant as a multilinear function of rows square matrices A, cofactor expansion and proof of det A^T=det A; Theorem: Determinant of an upper-triangular matrix is the product of its diagonal entries; Theorem: If rows of A are linearly-dependent, det A=0; Transformation rule of det A under elementary row operators; Theorem: A is invertible if and only if it has a nonzero determinant; Theorem: det (AB)=det A det B; Statement of Cramer’s Rule.	Pages 189-195 of Lay et al
17	Apr. 03	Cramer’s Rule and its application in the derivation of a formula for the inverse of an invertible matrix	Pages 195-198 of Lay et al
18	Apr. 05	Application of the determinant in computing areas and volumes: Rotation matrices have unit determinant, Rotation of columns of a matrix does not change its determinant, Areas of parallelograms in terms of the determinant of a matrix, effect of a linear transformation T:R^2->R^2 on the area of a parallelogram, regions obtained by the disjoint union of parallelograms, the limit of large number of small parallelogram, area of an ellipse, Extension to R^3: Volume of parallelepiped and the effect of linear transformations T:R^3->R^3 on it; Definition of an abstract real vector space	Pages 198-208 of Lay et al
Spring Break
19	Apr. 17	Review of the definition of a real vector space, Examples 1: Trivial vector space {0}, 2: R^n with component-wise addition and scalar multiplication, 3) R^{m x n}:= set of m x n matrices with usual addition and scalar multiplication of matrices, 4) C(I):= set of functions f:I->R with domain I which is an interval of real numbers (detailed proof of all the conditions for a vector space for this example); three examples of sets with two binary operations that do not form a vector space; Subspaces of a vector space V: Trivial subspaces ({0} and V); Subspace of upper-triangular matrices in R^{2x2}; Subspace of polynomials in C(I); Subspaces of n-times differentiable functions in C(I).	Pages 208-211 of Lay et al
20	Apr. 19	Some basic consequences of the axioms of a vector space: Uniqueness of the zero vector and –v for each v, proof of 0v=0, (-1)v=-v, every subspace is a vector space; linear combination, span, spanning subsets, finite- and infinite-dimensional vector spaces, vector space of polynomials of degree at most n is finite dimensional, vector space of all polynomials p:R->R is infinite-dimensional	Pages 211-213 of Lay et al
21	Apr. 24	Theorem: Span of A if the smallest subspace containing A, linearly-independent subsets and bases of a vector space, basis for space of polynomials of degree not larger than n, basis for the space of all polynomials	Pages 226-234 of Lay et al
22	Apr. 26	Linear transformations T:V->W mapping a vector space V to a vector space W; Examples, Theorem: T(zero)=zero, Null space of T, Theorem: Null space and range of T are subspaces of V and W, respectively, Null space of a differential operator, Theorem: T is 1-to-1 if and onlf if its null space is trivial, vector space isomorphisms, examples	Pages 221-223 of Lay et al
23	May 03	Coordinates and the coordinate vector with respect to a basis, the ismorphism mapping vectors to their coordinate vector in a basis, Theorem: If V is a vector space, B is a basis of V with n elements, and A is a subset of V with m>n elements, then A is linearly-dependent, Theorem: In finite-dimensional vector space all bases have the same number of elements, Theorem: Subspaces of a finite-dimensional vector space V are finite-dimensional and their dimension cannot exceed the dimension of V, Theorem: Every finite-dimensional vector space has a basis.	Pages 234-248 of Lay et al
Midterm Exam 2
24	May 08	The isomorphism that implements a change of basis, the matrix connecting the coordinate vector in one basis to another; The matrix representation of a linear transformation T:V->V for a finite-dimensional vector space; Are there bases yielding a diagonal matrix representation?	Pages 257-262 & 307-309 of Lay et al
25	May 10	Effect of a basis transformation on the matrix representation of a linear transformation T:V->V, the similarity transformations and diagonalizable transformations. Eigenvalue problem for matrices: The characteristic equation, and the determination of the eigenvectors; Example: A 2x2 matrix with two distinct real eigenvalues	Pages 309-311, 284-287 & 294-295 of Lay et al
26	May 15	Diagonalizable matrices and their diagonalization: Theorem: If an nxn matrix has n distinct eigenvalues with eigenvectors v1, v2, …, vn, then {v1, v2, …, vn} is linearly independent. Theorem: If an nxn matrix A has n eigenvectors forming a linearly-independent subset, then there is a similarity transformation that diagonalizes A. Example of a non-diagonalizable matrix; Example of a diagonalizable matrix with repeated eigenvalues, Example of a diagonalizable matrix with complex eigenvalues, complex numbers and their basic algebraic properties, complex matrices	Pages 288, 299-306 & 313-319 of Lay et al
27	May 17	Euclidean Inner product and Orthogonality: The dot (Euclidean inner) product on R^n and its basic properties, orthogonal and unit vectors in R^n, Pythagorean theorem, orthogonal subsets of R^n, Orthogonal complement of a subspace of R^n, Theorem: A finite orthogonal subset of R^n that does not contain the zero vector is linearly independent. Orthogonal and orthonormal bases of subspaces of R^n, the coordinate of a vector in orthogonal and orthonormal bases. The construction of an orthonormal basis of a given subspace of R^n (the Gram-Schmidt Process)	Pages 348-352, 356-358 & 372-374 of Lay et al

Note: The pages from the textbook listed above may not include some of the material covered in the lectures.