Introductory Linear ALgebra -- Lecture Notes

Math 220 (Fall 2013): Topics covered, and Lecture notes in Introductory Linear Algerbra

Lecture capture videos were originally made using Tegrity, and later migrated to Panopto.

Scribes from all lectures (as a single big file)

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

Scribes from all lectures (as a single big file)