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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 13th (Week 1)
Tuesday, January 14th
Topics: Course logistics; systems of linear equations; geometry and size of the solution set; (in)consistent systems; parametric form; the augmented and coefficient matrices; elementary row operations; equivalent systems.
Reading: §1.2 Systems; §1.3 Gaussian Elimination.
Videos:
Thursday, January 16th
Topics: Elementary row operations (contd); equivalent systems (contd); echelon form; Gaussian elimination; reduced-row echelon form; pivots and rank; cases for the solution set.
Reading: §1.3 Gaussian Elimination.
Videos:
Week of January 20th (Week 2)
Tuesday, January 21st
Topics: The space \(M_{m\times n}\); matrix arithmetic (addition, subtraction, scalar multiplication); the vector space \(\mathbb{R}^n\); abstract vector spaces; the spaces \(P(\mathbb{R})\) and \(P_n(\mathbb{R})\).
Reading: §1.1 The Vector Space of \(m\times n\) Matrices.
Videos:
Thursday, January 23rd
Topics: The function spaces (\(\mathcal{F}\), \(\mathcal{C}\), \(\mathcal{D}\)); linear combinations; linear (in)dependence; formally checking if \(v \in \mathrm{Span}\{v_1,\cdots,v_k\}\) by solving a linear system.
Reading: §1.3 Gaussian Elimination.
Videos:
Week of January 27th (Week 3)
I will be out of town this week; please see below for necessary adjustments.
HW 2 is due on Wednesday, February 5th at 11 PM (note: unusual day!).
No Office Hours this week (in view of my travel).
Tuesday, January 28th
Thursday, January 30th
Online Class (via Zoom).
Topics: Linear independence (contd); span; geometric aspects of independence and span in \(\mathbb{R}^n\).
Reading: §1.2 Systems; §2.1 The Test for Linear Independence.
Notes: (Morning, Evening)
Video:
Week of February 3rd (Week 4)
HW 2 is due on Wednesday, February 5th at 11 PM (note: unusual day!).
HW 3 is due on Saturday, February 8th at 11 PM.
Tuesday, February 4th
Topics: Testing if a set is linearly independent; pivot vectors and columns; the connection between linear (in)dependence in \(\mathbb{R}^m\) and pivot vectors; transpose and symmetric matrices.
Reading: §1.2 Systems; §2.1 The Test for Linear Independence.
Video:
Note: Due to a technical issue with the webcam, the Morning06 recording only begins halfway through class. I recommend watching the Afternoon06 recording instead.
Thursday, February 6th
Topics: Review of span; 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 and row space of a matrix.
Reading: §1.4 Column Space and Nullspace; §2.1 The Test for Linear Independence.
Video:
Week of February 10th (Week 5)
Tuesday, February 11th
Topics: The nullspace of a matrix; the linearity of matrix multiplication; describing \(\textrm{Null}(A)\) as a span; the translation theorem.
Reading: §1.4 Column Space and Nullspace; §2.1 The Test for Linear Independence.
Video:
Thursday, February 13th
Topics: Equivalent characterizations of subspaces; basis; standard bases of \(\mathbb{R}^n\), \(P_n(\mathbb{R})\), and \(M_{m\times n}(\mathbb{R})\); Computing bases of column spaces; computing bases of subspaces of \(\mathbb{R}^n\); linear relations between columns of a matrix are preserved under EROs; intuitive notion of dimension; formal definition of dimension; \(\dim V = |\mathcal{B}|\) where \(\mathcal{B}\) is any basis of \(V\).
Reading: §2.2 Dimension.
Video:
Note: Due to a technical issue, the Morning09 lecture was not recorded. Please watch the Afternoon09 lecture.
Week of February 17th (Week 6)
Tuesday, February 18th
Topics: Review of computing bases of column spaces and nullspaces; computing bases of row spaces; \(A\) and \(B\) are row-equivalent implies they have the same row space; 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.
Reading: §2.2 Dimension; §2.3 Row Space and the rank-nullity theorem.
Video:
Thursday, February 20th
Topics: Any two of \(|S| = \dim V\), \(S\) is spanning, and \(S\) is linearly independent implies the third; rank as dimension; 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; nullity; 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; properties of nonsingular matrices.
Reading: §2.3 Row Space and the rank-nullity theorem.
Video:
Week of February 24th (Week 7)
Midterm 1 is on Thursday, February 27th from 8:00 PM to 9:00 PM in BHEE 170.
HW 6 is due on Wednesday, March 5th at 11 PM (note: non-standard day!).
In view of Midterm 1, Office Hours will be on T, 2:00 PM to 4:00 PM and Th, 3:30 to 4:30 PM this week.
Tuesday, February 25th
Thursday, February 27th
Topics: The product \(A \vec{x}\) as matrix multiplication; the index formula for matrix multiplication; the identity matrix; 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; invertible matrices; the inverse of a matrix; the equivalence between invertibility, nonsingularity, and full-rank; the inverse of a \(2 \times 2\) matrix.
Reading: §3.2 Matrix Multiplication (Composition); §3.3 Inverses.
Video:
Week of March 3rd (Week 8)
HW 6 is due on Wednesday, March 5th at 11 PM (note: non-standard day!).
HW 7 is due on Friday, March 14th at 11 PM (note: non-standard day!).
Tuesday, March 4th
Topics: The algorithm for computing inverses using EROs; proof of correctness of the algorithm; the uniqueness of inverses; the inverse of the transpose is the transpose of the inverse; the inverse of a product; the rank of \(AB\) is at most the rank of \(A\) (resp. \(B\)); transformations; the identity transformation; matrix transformations.
Reading: §3.1 The Linearity Properties; §3.2 Matrix Multiplication (Composition); §3.3 Inverses.
Video:
Thursday, March 6th
Topics: Rotations in \(\mathbb{R}^2\) as a matrix transformation; domain and codomain; equality of transformations; linear transformations; examples of linear transformations; matrix transformations as linear transformations; every linear transformation of Euclidean spaces is a matrix transformation; formula for the matrix of a linear transformation; image of a set under a transformation; the image of a line segment under a linear transformation; differentiation and integration as linear maps between function spaces.
Reading: §3.1 The Linearity Properties; §3.5 The Matrix of a Linear Transformation.
Video:
Week of March 10th (Week 9)
HW 7 is due on Friday, March 14th at 11 PM (note: non-standard day!).
Office Hours will be on Th, 3:30 PM to 6:30 PM this week (no office hours on Tuesday).
Tuesday, March 11th
Topics: 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 \(\mathbb{R}^n\); 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.
Reading: §3.1 The Linearity Properties; §3.3 Inverses; §3.5 The Matrix of a Linear Transformation.
Video:
Thursday, March 13th
Topics: 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; 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 \(\mathbb{R}^2\); the coordinate transformation \(C_{\mathcal{B}}\).
Reading: §3.5 The Matrix of a Linear Transformation.
Video:
Week of March 17th (Week 10)
Tuesday, March 18th
Thursday, March 20th
Week of March 24th (Week 11)
HW 8 and HW 9 are due on Wednesday, April 2nd at 11 PM (note: unusual day!).
Office Hours will be on T, 2:00 PM to 3:30 PM this week (no office hours on Thursday).
Tuesday, March 25th
Topics: The coordinate vector \([\vec{x}]_{\mathcal{B}}\) (contd); \(P_{\mathcal{B}}\) and \(C_{\mathcal{B}}\); \(P_{\mathcal{B}}^{-1} = C_{\mathcal{B}}\); the point transformation and the coordinate transformation are isomorphisms; the matrix of a linear transformation of general vector spaces; change of basis formula for linear transformations of Euclidean spaces; a motivating example for change of basis; determinants as a criterion for invertibility.
Reading: §3.5 The Matrix of a Linear Transformation; §4.1 Definition of Determinant.
Video:
Thursday, March 27th
Topics: Minors and cofactors; the recursive definition of determinants; examples of using the definition to compute determinants; 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; sufficient conditions for \(\det A = 0\); the action of row addition on determinants; using EROs to compute determinants; upper and lower triangular matrices.
Reading: §4.1 Definition of Determinant; §4.2 Reduction and Determinant; §4.3 Inverses.
Video:
Week of March 31st (Week 12)
HW 8 and HW 9 are due on Wednesday, April 2nd at 11 PM (note: unusual day!).
HW 10 is due on Saturday, April 5th at 11 PM.
Extra Office Hours at M, 3:30 PM to 5:00 PM this week (in addition to usual Office Hours) in view of the postponed homework.
Tuesday, April 1st
Topics: The determinant of a triangular matrix; row equivalence and determinant; the determinants of RREF matrices; the connection between determinant, rank, and invertibility; Cramer's rule; the adjugate or adjoint of a matrix; formula for the inverse in terms of adjugates and determinants; uniqueness of the determinant as an alternating multilinear form; \(\det(AB) = (\det A)(\det B)\); the geometric interpretation of determinant.
Reading: §4.2 Reduction and Determinant; §4.3 Inverses.
Video:
Thursday, April 3rd
Topics: The geometric interpretation of determinant; the determinant as the scaling factor of a transformation on areas & volumes; eigenvectors and eigenvalues; using eigenvectors and basis change to compute \(A^k \vec{v}\) for large \(k\); the characteristic polynomial \(\det(A-\lambda I)\) of a matrix; computing eigenvalues; computing eigenvectors; eigenspaces.
Reading: §4.1 Definition of Determinant; §5.1 Eigenvectors.
Video:
Week of April 7th (Week 13)
Midterm 2 is on Wednesday, April 9th from 8:00 PM to 9:00 PM in BHEE 170.
HW 11 is due on Saturday, April 12th at 11 PM.
Tuesday, April 8th
Thursday, April 10th
Topics: Issues relating to nonexistence of eigenvalues or eigenvectors; algebraic multiplicity \(n_j\) and geometric multiplicity \(m_j\) of an eigenvalue \(\lambda_j\); diagonalizable and nondiagonalizable matrices; diagonalization; diagonalization as a change of basis; eigenvectors of distinct eigenvalues are linearly independent; 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.
Reading: §5.2 Diagonalization; §5.3 Complex eigenvectors.
Video:
Week of April 14th (Week 14)
I will be out of town for part of this week; please see below for necessary adjustments.
Office Hours will be on Th, 2:00 PM to 5:00 PM this week (no office hours on Tuesday).
HW 12 is due on Saturday, April 19th at 11 PM.
Tuesday, April 15th
Thursday, April 17th
Week of April 21st (Week 15)
Tuesday, April 22nd
Thursday, April 24th
Week of April 28th (Week 16)
No HW due this week (in view of Quiet Period).
Extra Office Hours at F, 2:00 PM to 4:00 PM this week (in addition to usual Office Hours) in view of the final exam.
Tuesday, April 29th
Thursday, May 1st
Week of May 5th (Final Exam Week)
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