We will be using sage as a symbolic calculator.
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 2021 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.
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.
Exercises on Determinants
Four multiple choice practice questions on permutations, determinants,
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.
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,
- 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...