Math 220 (Fall 2013): Topics covered, and Lecture notes in Introductory Linear Algerbra
Lec # | Date | Topic(s) | Notes | Tegrity |
---|---|---|---|---|
1 | Aug 20 | syllabus, intro to mymathlab.com, 2D example (2 linear equations in 2 variables), graphical solution | scribe | video |
2 | Aug 22 | Gaussian elimination, augmented matrix, elementary row operations (EROs), \( \begin{bmatrix} 0 & 0 & \cdots & 0 & | & {\bf \star}\neq 0 \end{bmatrix}\) -- inconsistent system | scribe | video |
3 | Aug 27 | nonzero row, leading entry, echelon form and reduced echelon form, row reduction | scribe | video |
4 | Aug 29 | symbolic notation for echelon form, basic and free variables, parametric form of solutions, vectors, linear combination, span, | scribe | video |
5 | Sep 3 | pictures of span in 2D & 3D, span all of \( \mathbb{R}^n \), matrix form \( A \mathbf{x} = \mathbf{b} \) | scribe | video |
6 | Sep 5 | \( A \mathbf{x} = \mathbf{b} \) consistent \( \forall \mathbf{b}\) iff \(A\) has a pivot in every row, homogenous system, trivial solution, parametric vector form of solutions | scribe | video |
7 | Sep 10 | Solutions to \(A\mathbf{x} = \mathbf{b}\) in relation to those of \(A\mathbf{x} = \mathbf{0}\), linear independence (LI) of vectors, special case of LI for a single vector | scribe | video |
8 | Sep 12 | LI of two vectors, with \( \mathbf{0} \), \(\{\mathbf{v}_1,\dots,\mathbf{v}_n\}\) is LD when \( \mathbf{v}_i \in \mathbb{R}^m\) and \(n > m\), (matrix) transformations, (co)domain, image, range | scribe | video |
9 | Sep 17 | review of problems from Sections 1.4, 1.5, and 1.7 (as we are slowing down:-) | scribe | video |
10 | Sep 19 | linear transformations (LT), preserve vector addition and scalar multiplication, matrix of an LT | scribe | video |
11 | Sep 24 | matrix of an LT, geometric transformations - rotation, reflection, shear, projection, onto and one-to-one transformations | scribe | oops:-( |
12 | Sep 26 | problems on matrix of LT, onto and one-to-one LTs, onto iff pivot in every row, 1-to-1 iff pivot in every column | scribe | video |
13 | Oct 1 | review of practice midterm exam | scribe | video |
14 | Oct 3 | midterm | exam | |
15 | Oct 8 | class canceled | ||
16 | Oct 10 | By Prof. McDonald: matrix multiplication, power and transpose of a matrix; notes from Prof. McDonald | scribe | NA |
17 | Oct 15 | inverse of a matrix, inverse of \(2 \times 2 \) matrix, determinant, properties of matrix inverses | scribe | video |
18 | Oct 17 | inverting \( n \times n \) matrix, \( [ A | I] \longrightarrow [ I | A^{-1} ]\), using properties of inverses, invertible matrix theorem (IMT) | scribe | video |
19 | Oct 22 | invertible matrix theorem (IMT) - statements (a)-(l), problems using IMT | scribe | video |
20 | Oct 24 | subspaces - of \( \mathbb{R}^2 \), generated by \(\{ \mathbf{v_1},\dots,\mathbf{v_p}\}\); column and null spaces of \(A\) - \( \operatorname{Col} A, \operatorname{Nul} A\), basis for a subspace | scribe | video |
21 | Oct 29 | By Prof. McDonald: bases for \( \operatorname{Col} A\),\( \operatorname{Nul} A\), dimension of a subspace, rank, rank theorem, IMT (contd..); Prof. McDonald's notes | scribe | NA |
22 | Oct 31 | class canceled | ||
23 | Nov 5 | coordinates of \({\bf x}\) in a basis \(\mathcal{B}\): \( [{\bf x}]_{\mathcal{B}} \), more on dimension and rank, determinant of a \( 3 \times 3\) matrix, expand along Row 1 | scribe | video |
24 | Nov 7 | Computer project, intro to MATLAB, \(n \times n\) determinant by expanding along Row 1, cofactor expansion along any row/column | scribe | video |
25 | Nov 12 | \(\operatorname{det}\) of triangular matrix, \(\operatorname{det}(A)\) using EROs, \(A^{-1}\) exists \(\Leftrightarrow \operatorname{det}(A) \neq 0\), \(\operatorname{det}(A^T) = \operatorname{det}(A)\), \(\operatorname{det}(AB) = \operatorname{det}(A)\operatorname{det}(B)\) | scribe | video |
26 | Nov 14 | \(\operatorname{det}(B^{-1}AB) = \operatorname{det}(A)\) when \(\operatorname{det}(B) \neq 0\), eigenvalues and eignevectors, \((A^T = A) \Rightarrow \) all eignvalues of \(A\) are real | scribe | oops:-( |
27 | Nov 19 | testing eigenvalues/eigenvectors, eigenspace, eigenvalues of triangular matrices are diagonal entries, all rows of \(A\) adding to \(s\) | scribe | video |
28 | Nov 21 | characteristic polynomial & equation, multiplicity of eigenvalue, similar matrices, EROs and eigenvalues, Matlab for project | scribe | video |
29 | Dec 3 | scalar (or dot) product of vectors, length of vector, orthogonal vectors and sets, orthogonal basis, orthogonal projection | scribe | video |
30 | Dec 5 | write vector \(\mathbf{y}\) as \(\mathbf{y_u} + \mathbf{v}\), where \(\mathbf{y_u} \in\) span\(\{\mathbf{u}\}\) and \(\mathbf{v} \perp \mathbf{y_u}\), review for final exam | scribe | video |