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An Introduction to Elementary Linear Algebra


We will be using sage as a symbolic calculator.

Zoom lecture Nov. 22. Passcode: Dt.&63y3

Exam4.pdf Due Tuesday Dec. 6 before 5pm.

Answer Form. Use this Google form to submit your answers.

Online Textbooks

Linear Algebra (in .pdf) Local copy.
Answers to the exercises (pdf)

Linear Algebra using Sage (in .pdf)


What was a Stem-and-Leaf again?

Table Of Contents

Fall 2022 Syllabus.
An Introduction to Geometric Algebra.
Just the basics in 3D.
Another intro for GA of 3D.
Another quick introduction.
Overview of Geometric Albegra
Hestenes Oersted Medal Lecture.
Hestenes Primer on GA
Colorful pictures and applications to physics.
Geometric Algebra Exercises.
Tangents to a circle
Detailed solution with GA in GeoGebra.
Systems of Linear Equations
Examples wih solutions using Maple.
Sage Quickref-linalg.pdf
Sage Quick Reference: Linear Algebra.
sage file: matrixops.sage
Gauss-Jordan elimination in sage, Reflexions, Rotations, Simplification.
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 step-by-step 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.
Vector Axioms
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.
Geometric Algebra in Maple. The Clifford Package
The package with instructions and examples.
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

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 anti-commutatitivity 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...
[Other Linear Algebra sites on the internet]
Carlos Rodriguez <>
Last modified: Tue Jun 4 12:38:10 EDT 2002