
We will be using sage as a symbolic calculator. We'll also borrow
Maple notation for displaying some formulas. For a quick introduction
to the basics of Maple read Maple: An Introduction (PostScript). A short (html) file describing
the use of Maple in linear algebra is Linear Algebra Using Maple by Victoria Bush.
Online Textbook
Linear Algebra by Jim Hefferon.
Linear Algebra (in .pdf) Local copy.
Answers to the exercises (pdf)
Question:
What was a StemandLeaf again?
Table Of Contents
 EXAM4: Take Home
 Slip under my office door (ES319B) before Fri. May 15 at 1:00PM
 Syllabus
 Spring 2015 Syllabus.
 Systems of Linear Equations
 Examples wih solutions using Maple.
 Sage: Elementary row operations: (.pdf), (.sws).
 Elementary row operations with sage
 Elementary Row Operations
 Transforming the system of linear equations to be able to solve it.
 Gaussian Elimination
 A stepbystep example using Maple on a 4 by 5 matrix.

Elementary Matrices and Inverses
 Another example of step by step gaussian elimination

A system with 0,1 or infinitely many solutions
 Example of a system that can have many kinds of solutions depending
on the values of two parameters.

Exercises for Exam1
 Seventeen (yes 17) problems on matrices and systems of linear equations.

Exercises on Determinants
 Four multiple choice practice questions on permutations, determinants,
Cramer's rule.
 Imaginary Numbers are not Real
 Html and Postscript versions available online.
 An Introduction to Geometric Algebra.

 Geometric Algebra in Maple. The Clifford Package
 The package with instructions and examples.
 compose_rotations
 A maple procedure to compute the axis and angle equivalent to the composition
of two rotations.
 Eigenvalues and Eigenvectors without determinants!
 Modern approach to diagonalization.
Introduction to the Algebra and Geometry of Euclidean Space
 Vectors
 Introduction to the concept of vector. Magnitud,
direction, addition.
 Vector Geometry
 Cartesian and spherical coordinate systems. Describing, surfaces,
lines, points with vectors.

Working with vectors in Maple
 Using maple to compute addition of vectors, magnitudes, angles.
The plane in the wind problem is here...
 The Dot Product
 Introducing the inner product. Scalar and vector projections.
 The Cross Product
 Definition. Cross products of the i,j,k basis vectors. Examples.

Properties of Cross products
 Maple proofs of the distributivity and anticommutatitivity
properties of cross products.

Cross products are NOT associative.
 Maple proof that cross products are not associative.

Applications of the cross product: planes, volumes
 Triple products. The volume generated by 3 vectors. Projected Area.

Lines with Maple
 Position vector plus t times the velocity vector: Howto with maple.

The plane through 3 points
 The equation of the plane containing 3 given points. The maple
procedure P3points for computing it is here...

The plane containing two lines
 The equation of the plane containing two given lines. The maple
procedure interlines for finding the point of intersection
of two lines in 3D is here...

The distance from a point to a line
 How far away is this point from that line?
The maple proc d2line is here...

The distance from a point to a plane
 How far is that point from this plane ?
The maple proc p2plane is here...

Plane containing two lines: Example1
 Given two lines in symmetric form, maple is used to find
the plane that contains them. A picture of the plane with the
two lines is here...

Example: angle of diagonals
 Simple Maple proof that when the diagonals of a rectangle
intersect at right angles then the rectangle is a square.

Example: bisecting the angle between u and v
 Length of u times v plus length of v times u does it!
The proof with maple is here...

Two planes and one point
 The equation of the plane that contains the line of intersection
of two other planes and a given point.

Two planes, angle, line..
 Finding the angle between two planes and the line of
intersection in symmetric form.
 A few review exercises
 Seven problems on lines, planes, angles, innerprods etc...
