# 2250 Lecture Record Week 5 F2009

**Last Modified**: October 02, 2009, 13:18 MDT. **Today**: March 23, 2018, 05:16 MDT.
### Week 5, Sep 21 to 25: Sections 3.2, 3.3, 3.4, 3.5

**21 Sep**: Frame Sequences. Three Possibilities. No matrices. Sections 3.2, 3.3.

Due today maple L1.1, L1.2, L1.3, L1.4

Lecture: 3.1, 3.2, 3.3, frame sequences, general solution, three possibilities.

Lead variable, Free variable, signal equation, echelon form and the last frame test.

A detailed account of the three possibilities.
**How to solve a linear system using the toolkit **
- Toolkit: swap, combo, mult are the elementary operations
- Toolkit operations neither create nor destroy solutions!
- Frame sequences
- The unique solution case,
- The no solution case, and
- The infinitely many solution case.
- Examples.

- Computer algebra systems and error-free frame sequences.
- How to program maple to make a frame sequence without errors.

**Problems 3.2**: Back-substitution should be presented as combo
operations in a frame sequence, not as isolated, incomplete algebraic jibberish.
**Problem 3.2-24:** See references on 3 possibilities with symbol k.

**Beamer slides**: 3 possibilities with symbol k (89.5 K, pdf, 05 Oct 2007)

**Slides**: 3 possibilities with symbol k (97.9 K, pdf, 25 Oct 2007)

**Manuscript**: Example 10 in Linear algebraic equations no matrices (292.7 K, pdf, 08 Mar 2009)

Prepare 3.3 problems 8, 18 for next time. Please use frame sequences
to display the solution, as in today's lecture examples. It will be a
sequence of augmented matrices. Yes, you may use maple to make the frame
sequence and to do the answer check [rref(A);].
**References for this lecture.**

**Slides**: Linear equations, reduced echelon, three rules (155.6 K, pdf, 06 Aug 2009)

**Slides**: Infinitely many solutions case (93.8 K, pdf, 03 Oct 2009)

**Slides**: No solution case (58.4 K, pdf, 03 Oct 2009)

**Slides**: Unique solution case (86.0 K, pdf, 03 Oct 2009)

**Manuscript**: Linear algebraic equations, no matrices (292.7 K, pdf, 08 Mar 2009)

**22 Sep**: Augmented Matrix for System Ax=b. RREF. Last Frame Algorithm. Sections 3.3, 3.4.

Review: Last frame test. The RREF of a matrix.

Review: Last frame algorithm.

Drill: last frame algorithm and the scalar form of the solution.

Lecture: 3.3 and 3.4.

Translation of equation models to (augmented) matrix models and back.

Combo, swap and multiply for matrix models.

Frame sequences for matrix models.

Special matrices: Zero matrix, identity matrix, diagonal matrix, upper and lower triangular matrices, square matrix.

Homogeneous system with a unique solution.

Homogeneous system with more variables than equations.

Equation ideas can be used on a matrix A. View matrix A
as the set of coefficients of a homogeneous linear system Ax=0. The augmented
matrix B for this homogeneous system would be the given matrix with a
column of zeros appended: B=aug(A,0).
## Last Frame Algorithm

How to use maple to compute a frame sequence. Example is Exercise 3.2-14 from Edwards-Penney.

**Maple**: Frame Sequence in maple, Exercise 3.2-14 (3.1 K, mws, 23 Sep 2009)

**Maple**: Frame Sequence in maple, Exercise 3.2-14 (2.8 K, txt, 23 Sep 2009)
Answer checks should also use the online FAQ.

**html**: Problem notes F2009 (4.0 K, html, 22 Sep 2009)
**23 Sep**: Matrix Operations. Frame Sequence Analysis for Matrices. Section 3.4, 3.5.

Due today, Page 152, 3.1: 6, 16, 26

Review: Answer checks with matlab, maple and mathematica. Pitfalls.

Review of the three possibilities and frame sequence analysis to find the general solution.
Frame sequences with symbol k.

**Slides**: Three possibilities, theorems on infinitely many solutions, equations with symbols (100.3 K, pdf, 23 Sep 2009)

**Beamer slides**: 3 possibilities with symbol k (89.5 K, pdf, 05 Oct 2007)

**Slides**: 3 possibilities with symbol k (97.9 K, pdf, 25 Oct 2007)

**Manuscript**: Example 10 in Linear algebraic equations no matrices (292.7 K, pdf, 08 Mar 2009)

Lecture: Matrix. Vector. Matrix multiply, college algebra, examples.
Matrix rules [vector space rules]. Matrix multiply rules.

**Manuscript**: Vectors and Matrices (266.8 K, pdf, 09 Aug 2009)

**Manuscript**: Matrix Equations (162.6 K, pdf, 09 Aug 2009)

Examples: how to multiply matrices on paper.

**Slides**: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)

**Maple:** Lab 2 problem L2.1 discussed today. Solution projected for L2.1.

**24 Sep**: Fusi and Richins
Exam 1 review, problems 1,2,3,4,5. Maple lab 2 details. Review of Ch3 problems.
**25 Sep**: General solution of Ax=b. Section 3.5.

**Review and drill on linear equations: **
- Unique solution case,
- infinitely many solution case and the last frame algorithm.
- Equality of vectors. Convert vector equations <==> scalar equations.
- Matrix-vector equations Ax=b.
- Vector form of a solution
in both the unique solution case and the last frame algorithm case.
- Superposition.

General solution X=X_{0}+t_{1} X_{1} + t_{2} X_{2} + ... + t_{n} X_{n}.

Lecture: General structure of linear systems. Superposition.
Matrix formulation Ax=b of a linear system
- Properties of matrices: addition, scalar multiply.
- Matrix multiply rules. Matrix multiply Ax for x a vector.
- Linear systems as the matrix equation Ax=b.

## References for Chapters 3 and 4

**Slides**: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)

**Manuscript**: Linear equations, reduced echelon, three rules (45.8 K, pdf, 22 Sep 2006)

**Manuscript**: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)

**Manuscript**: Linear algebraic equations, no matrices (292.7 K, pdf, 08 Mar 2009)

**Manuscript**: Vectors and Matrices (266.8 K, pdf, 09 Aug 2009)

**Manuscript**: Matrix Equations (162.6 K, pdf, 09 Aug 2009)

**Transparencies**: Ch3 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)

**Slides**: Elementary matrix theorems (114.4 K, pdf, 03 Oct 2009)

**Slides**: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)

**Slides**: Linear equations, reduced echelon, three rules (155.6 K, pdf, 06 Aug 2009)

**Slides**: Infinitely many solutions case (93.8 K, pdf, 03 Oct 2009)

**Slides**: No solution case (58.4 K, pdf, 03 Oct 2009)

**Slides**: Unique solution case (86.0 K, pdf, 03 Oct 2009)

**Maple**: Lab 5, Linear algebra (94.3 K, pdf, 19 Jul 2009)

**html**: Problem notes F2009 (4.0 K, html, 22 Sep 2009)

**Slides**: Determinants 2008 (167.7 K, pdf, 03 Oct 2009)

**Manuscript**: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)

**Slides**: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)

**Slides**: Digital photos, Maxwell's RGB separations, visualization of matrix add (153.7 K, pdf, 16 Oct 2009)

**Slides**: Inverse matrix, frame sequence method (71.6 K, pdf, 02 Oct 2009)

**Slides**: More on digital photos, checkerboard analogy (109.5 K, pdf, 02 Oct 2009)

**Slides**: Rank, nullity and elimination (111.6 K, pdf, 29 Sep 2009)

**Slides**: Base atom, atom, basis for linear DE (85.4 K, pdf, 20 Oct 2009)

**Slides**: Orthogonality (87.2 K, pdf, 10 Mar 2008)

**Slides**: Partial fraction theory (121.5 K, pdf, 30 Aug 2009)

**Slides**: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)

**Text**: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)

**Text**: History of telecom companies (1.1 K, txt, 05 Oct 2008)