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Schedule
Notes
The schedule is tentative and will change as the course progresses.
For information about add/drop dates and other matters related to course registration, please consult the Registrar's Website.
All dates and times are in ET (Eastern Time).
Section numbers are as in Linear Algebra, Ideas and Applications (4th Edition) by Richard C. Penney.
There will usually be one homework assignment due on Saturday every week.
Week of January 8th (Week 1)
Monday, January 8th
Wednesday, January 10th
Friday, January 12th
Topics: Gaussian elimination (contd); pivots and rank (contd); cases for the solution set; reduced-row echelon form; the space \(M_{m\times n}\); matrix arithmetic (addition, subtraction, scalar multiplication); the vector space \(\mathbb{R}^n\).
Reading: §1.3 Gaussian Elimination; §1.1 The Vector Space of \(m\times n\) Matrices.
Video: (Kaltura03)
Week of January 15th (Week 2)
HW 1 is now due on Saturday, January 20th at 11 PM (earlier due Friday, January 19th).
Office Hours will be on WF, 2:30 PM to 4:00 PM this week (in view of MLK day).
Monday, January 15th
Wednesday, January 17th
Topics: The vector space \(\mathbb{R}^n\); the spaces \(P(\mathbb{R})\), \(P_n(\mathbb{R})\) and \(M_{m\times n}(\mathbb{R})\); function spaces (\(\mathcal{F}\), \(\mathcal{C}\), \(\mathcal{D}^k\)); abstract vector spaces.
Reading: §1.1 The Vector Space of \(m\times n\) Matrices.
Video: (Kaltura04)
Friday, January 19th
Topics: Linear combinations; linear (in)dependence; span; geometric aspects of independence and span in \(\mathbb{R}^n\); formally checking if \(v \in \mathrm{Span}\{v_1,\cdots,v_k\}\) by solving a linear system.
Reading: §1.1 The Vector Space of \(m\times n\) Matrices; §1.3 Gaussian Elimination.
Video: (Kaltura05)
Week of January 22nd (Week 3)
I will be out of town this week; please see below for necessary adjustments.
HW 2 is due on Tuesday, January 30th at 11 PM (note: unusual day!).
No Office Hours this week (in view of my travel).
Monday, January 22nd
Wednesday, January 24th
Online Class (via Zoom).
Topics: Testing if a set is linearly independent; pivot vectors and columns; the connection between linear independence in \(\mathbb{R}^m\) and pivot vectors.
Reading: §1.2 Systems; §2.1 The Test for Linear Independence.
Video: (Kaltura06)
Notes: (Jan 24th)
Friday, January 26th
Week of January 29th (Week 4)
HW 2 is due on Tuesday, January 30th at 11 PM (note: unusual day!).
HW 3 is due on Saturday, February 3rd at 11 PM (earlier due Friday, February 2nd).
Monday, January 29th
Topics: Subspaces; multiplying an \(m\times n\) matrix with a vector in \(\mathbb{R}^n\); the equivalence between \(b\in \mathrm{Span}\{a_1,\cdots,a_n\}\) and solving the system \([ a_1 \cdots a_n | b]\); span as a subspace; the column space of a matrix.
Reading: §1.4 Column Space and Nullspace.
Video: (Kaltura08)
Wednesday, January 31st
Topics: Review of column spaces; the row space of a matrix; homogenous systems; the nullspace of a matrix; the linearity of matrix multiplication; describing \(\textrm{Null}(A)\) as a span.
Reading: §1.4 Column Space and Nullspace.
Video: (Kaltura09)
Friday, February 2nd
Topics: Review of nullspaces; the translation theorem; equivalent characterizations of subspaces; basis; standard bases of \(\mathbb{R}^n\), \(P_n(\mathbb{R})\), and \(M_{m\times n}(\mathbb{R})\).
Reading: §1.4 Column Space and Nullspace; §2.1 The Test for Linear Independence.
Video: (Kaltura10)
Week of February 5th (Week 5)
Monday, February 5th
Wednesday, February 7th
Topics: Proof of \(\dim V = |\mathcal{B}|\) where \(\mathcal{B}\) is any basis of \(V\); finite dimensional vector spaces; \(\mathbb{R}^\infty\) as an example of an infinite dimensional vector space; any two of \(|S| = \dim V\), \(S\) is spanning, and \(S\) is linearly independent implies the third.
Reading: §2.2 Dimension.
Video: (Kaltura12)
Friday, February 9th
Topics: Rank as dimension; \(A\) and \(B\) are row-equivalent implies they have the same row space; computing bases of row spaces; another way to compute bases of subspaces of \(\mathbb{R}^n\); row-rank = column-rank; column space as image of a matrix map; solvability of \(A\vec{x} = \vec{b}\) for every \(\vec{b} \in \mathbb{R}^m\) in terms of rank.
Reading: §2.3 Row Space and the rank-nullity theorem.
Video: (Kaltura13)
Week of February 12th (Week 6)
I will be out of town for part of this week; please see below for necessary adjustments.
HW 5 is due on Saturday, February 17th at 11 PM.
Office Hours will be on ThF, 2:30 PM to 4:00 PM this week.
Monday, February 12th
Online Class (via Zoom).
Topics: Nullity; computing bases of a nullspace; nullity as a measure of size of the solution set of \(A\vec{x} = \vec{b}\); the rank-nullity theorem; implications of the rank-nullity theorem; nonsingular matrices.
Reading: §2.3 Row Space and the rank-nullity theorem.
Video: (Kaltura14)
Notes: (Feb 12th)
Wednesday, February 14th
Topics: Properties of nonsingular matrices; matrix multiplication; the product \(A \vec{x}\) as matrix multiplication; the index formula for matrix multiplication; the identity matrix.
Reading: §2.3 Row Space and the rank-nullity theorem; §3.2 Matrix Multiplication (Composition).
Video: (Kaltura15)
Friday, February 16th
Week of February 19th (Week 7)
Midterm 1 is on Tuesday, February 20th from 8:00 PM to 9:00 PM.
HW 6 is due on Saturday, February 24th at 11 PM.
Monday, February 19th
Topics: The identity matrix (contd); the matrix product \(AB\) when \(B\) is a collection of columns; the matrix product \(AB\) when \(A\) is a collection of rows; the transpose of a product; algebraic properties of matrix multiplication; the rank of \(AB\) is at most the rank of \(A\) (resp. \(B\)); invertible matrices; the inverse of a matrix.
Reading: §3.2 Matrix Multiplication (Composition); §3.3 Inverses.
Video: (Kaltura17)
Wednesday, February 21st
Topics: The inverse of a \(2 \times 2\) matrix; the algorithm for computing inverses using EROs; proof of correctness of the algorithm; the equivalence between invertibility, nonsingularity, and full-rank; the uniqueness of inverses; the inverse of the transpose is the transpose of the inverse; the inverse of a product.
Reading: §3.3 Inverses.
Video: (Kaltura18)
Friday, February 23rd
Topics: Transformations; the identity transformation; matrix transformations; rotations in \(\mathbb{R}^2\) as a matrix transformation; domain and codomain; equality of transformations; image of a set under a transformation; the image of a line segment under a matrix transformation; linear transformations; examples of linear transformations; matrix transformations as linear transformations.
Reading: §3.1 The Linearity Properties; §3.2 Matrix Multiplication (Composition); §3.3 Inverses.
Video: (Kaltura19)
Week of February 26th (Week 8)
Monday, February 26th
Topics: Linear transformations (contd); every linear transformation of Euclidean spaces is a matrix transformation; differentiation and integration as linear maps between function spaces; composition of transformations; composition of linear transformations; matrix multiplication as composition of linear transformations between Euclidean spaces; the identity matrix \(I_n\) as the identity transformation of \(\mathbb{R}^n\).
Reading: §3.1 The Linearity Properties; §3.2 Matrix Multiplication (Composition).
Video: (Kaltura20)
Wednesday, February 28th
Topics: Invertible transformations; the inverse of a transformation; the inverse of a linear transformation is linear; the connection between the inverse matrix and the inverse of a linear transformation; ordered bases; uniqueness of representations in an ordered basis \(\mathcal{B}\); the coordinate vector \([\vec{x}]_{\mathcal{B}}\); Cartesian coordinates as the coordinates with respect to the standard basis; the geometric interpretation of a coordinate system in \(\mathbb{R}^2\).
Reading: §3.1 The Linearity Properties; §3.3 Inverses.
Video: (Kaltura21)
Friday, March 1st
Topics: The coordinate vector \([\vec{x}]_{\mathcal{B}}\) (contd); the geometric interpretation of a coordinate system in \(\mathbb{R}^2\) (contd); the point matrix \(P_{\mathcal{B}}\); the coordinate matrix \(C_{\mathcal{B}}\); \(P_{\mathcal{B}}^{-1} = C_{\mathcal{B}}\); a motivating example for change of basis; the matrix of a linear transformation of general vector spaces; change of basis formula for linear transformations of Euclidean spaces.
Reading: §3.5 The Matrix of a Linear Transformation.
Video: (Kaltura22)
Week of March 4th (Week 9)
I will be out of town for part of this week; please see below for necessary adjustments.
HW 8 is due on Saturday, March 23rd at 11 PM.
Office Hours will be on M, 2:30 PM to 4:00 PM this week (no office hours on Friday).
Monday, March 4th
Topics: The coordinate transformation \(C_{\mathcal{B}}\) and the point transformation \(P_{\mathcal{B}}\) for general vector spaces; the matrix of a linear transformation of general vector spaces (contd); injective (one-one) transformations; surjective (onto) transformations; bijective transformations; the kernel of a linear transformation; characterization of injectivity in terms of kernel; characterization of surjectivity in terms of image; isomorphism; isomorphic vector spaces have the same dimension; the point transformation and the coordinate transformation are isomorphisms; characterizations of isomorphisms between vector spaces of the same dimension.
Reading: §3.5 The Matrix of a Linear Transformation.
Video: (Kaltura23)
Wednesday, March 6th
Topics: Formula for the matrix of a linear transformation; characterizations of isomorphisms between vector spaces of the same dimension (contd); determinants as a criterion for invertibility; minors and cofactors; the recursive definition of determinants; examples of using the definition to compute determinants.
Reading: §3.5 The Matrix of a Linear Transformation; §4.1 Definition of Determinant.
Video: (Kaltura24)
Friday, March 8th
Week of March 11th (Week 10)
Monday, March 11th
Wednesday, March 13th
Friday, March 15th
Week of March 18th (Week 11)
Monday, March 18th
Topics: Review of the recursive definition of determinants; the geometric interpretation of determinant; the connection between rank and determinant for \(2 \times 2\) matrices; properties of determinants; \(\det A = \det A^{T}\); the action of row interchange or row scaling on determinants; additivity of determinants; multilinearity of the determinant; cofactor expansions along arbitrary rows or columns; .
Reading: §4.1 Definition of Determinant.
Video: (Kaltura25)
Wednesday, March 20th
Topics: Properties of determinants (contd); the action of row addition on determinants; sufficient conditions for \(\det A = 0\); upper and lower triangular matrices; the determinant of a triangular matrix; using EROs to compute determinants.
Reading: §4.1 Definition of Determinant; §4.2 Reduction and Determinant.
Video: (Kaltura26)
Friday, March 22nd
Topics: Review of the action of EROs on determinants; row equivalence and determinant; the determinants of RREF matrices; the connection between determinant, rank, and invertibility; uniqueness of the determinant as an alternating multilinear form; \(\det(AB) = (\det A)(\det B)\); the geometric interpretation of determinant; the determinant as the scaling factor in area & volume; the adjugate of a matrix.
Reading: §4.2 Reduction and Determinant; §4.3 Inverses.
Video: (Kaltura27)
Week of March 25th (Week 12)
Monday, March 25th
Topics: Cramer's rule; formula for the inverse in terms of adjugates and determinants; properties of determinants (contd); eigenvectors and eigenvalues; using eigenvectors and basis change to compute \(A^k \vec{v}\) for large \(k\); the connection to \(\det(A-\lambda I)\).
Reading: §4.3 Inverses; §5.1 Eigenvectors.
Video: (Kaltura28)
Wednesday, March 27th
Topics: The characteristic polynomial of a matrix; computing eigenvalues; computing eigenvectors; basic properties of eigenpairs; eigenspaces; issues relating to nonexistence of nonexistence of eigenvalues or eigenvectors.
Reading: §5.1 Eigenvectors.
Video: (Kaltura29)
Friday, March 29th
Week of April 1st (Week 13)
Midterm 2 is on Tuesday, April 2nd from 8:00 PM to 9:00 PM.
HW 11 is due on Saturday, April 6th at 11 PM.
Monday, April 1st
Topics: Computing eigenvalues and eigenvectors (contd); eigenbases; matrices with nonreal eigenvalues; diagonalizable and nondiagonalizable matrices; diagonalization; diagonalization as a change of basis; eigenvectors of distinct eigenvalues are linearly independent; a sufficient criterion for diagonalizability.
Reading: §5.1 Eigenvectors; §5.2 Diagonalization.
Video: (Kaltura31)
Wednesday, April 3rd
Topics: Diagonalization (contd); using diagonalization to compute \(A^k\); eigenvectors of distinct eigenvalues are linearly independent (contd); a sufficient criterion for diagonalizability (contd); algebraic multiplicity \(m_j\) and geometric multiplicity \(n_j\) of an eigenvalue \(\lambda_j\); \(n_j \leqslant m_j\); characterization of diagonalizability in terms of algebraic and geometric multiplicities; complex eigenvalues.
Reading: §5.2 Diagonalization.
Video: (Kaltura32)
Friday, April 5th
Topics: A matrix with no real eigenvalues; the complex numbers, \(\mathbb{C}\); arithmetic in \(\mathbb{C}\); the Argand plane; polar coordinates for the Argand plane; change of variable formulae for polar to rectangular and vice-versa; modulus and argument of a complex number; the geometric interpretations of addition and multiplication; diagonalizing a real matrix with complex eigenvalues; matrices and vectors with complex entries; EROs with \(\mathbb{C}\) as the field of scalars; the indistinguishability of \(i\) from \(-i\); de Moivre's formula.
Reading: §5.3 Complex eigenvectors.
Video: (Kaltura33)
Week of April 8th (Week 14)
HW 12 is due on Saturday, April 13th at 11 PM.
Office Hours will be on F, 2:30 PM to 4:00 PM this week (no office hours on Monday).
Monday, April 8th
Topics: Diagonalizing a real matrix with complex eigenvalues (contd); division of complex numbers; Hadamard's maxim; \(\mathbb{C}\) as a field; vector spaces over \(\mathbb{C}\); examples: \(\mathbb{C}^n\), \(P_n(\mathbb{C})\), \(M_{m\times n}(\mathbb{C})\); the fundamental theorem of algebra; all \(n\times n\) matrices over \(\mathbb{C}\) have \(n\) eigenvalues up to multiplicity.
Reading: §5.3 Complex eigenvectors.
Video: (Kaltura34)
Wednesday, April 10th
Topics: Validity of de Moivre's formula via geometry and Euler's formula; complex exponentiation via Euler's formula; Taylor series; trignometric angle formulae from complex exponentiation; type-2 matrices; the \(LU\) decomposition of a type-2 matrix; using \(LU\) decomposition to solve systems; multipliers.
Reading: §3.4 The \(LU\) factorization.
Video: (Kaltura35)
Friday, April 12th
Topics: Diagonalizing a matrix in \(M_n(\mathbb{C})\); complex matrices have a full set of eigenvalues; validity of \(n_j \leqslant m_j\) where \(n_j\) is the geometric multiplicity and \(m_j\) is the algebraic multiplicity by a change of basis; the characteristic polynomial does not depend on the coordinate system; complex matrices need not have a full set of eigenvectors; the golden ratio \(\varphi\); Fibonacci numbers; the limit of the ratio of Fibonacci numbers; the Fibonacci recurrence as the action of a matrix; diagonalizing the matrix.
Reading: §5.2 Diagonalization; §5.3 Complex eigenvectors; Lecture Notes.
Video: (Kaltura36)
Week of April 15th (Week 15)
Monday, April 15th
Wednesday, April 17th
Topics: Systems of ODEs; solving the one-dimensional equation \(x’ = a x\); coupled ODEs; decoupling via diagonalization; initial value problems; the ansatz \(\vec{x}(t) = e^{\lambda t} \vec{v}\); constant-coefficient linear \(n\)th order ODEs; systems of ODEs with complex eigenvalues; .
Reading: §5.2.2 Applications to Systems of Differential Equations.
Video: (Kaltura38)
Friday, April 19th
Topics: Dot product; norm; geometric interpretations; law of cosines; angles and distances in \(\mathbb{R}^n\) using dot products; Cauchy-Schwarz inequality; triangle inequality; orthogonal vectors; orthogonal sets; orthonormal sets; orthogonal bases; orthonormal bases; orthogonal sets are linearly independent; change of basis to an orthogonal basis; normalizing; projections.
Reading: §6 Orthogonality.
Video: (Kaltura39)
Week of April 22nd (Week 16)
Monday, April 22nd
Wednesday, April 24th
Friday, April 26th
Week of April 29th (Final Exam Week)
Final Exam is on Tuesday, April 30th from 7:00 PM to 9:00 PM.
Office Hours will be on MT, 2:30 PM to 4:00 PM this week.
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