Introductory examples.
Elementary Matrices, included in Linear Algebra WebNotes, contains information on matrices, together with proofs and examples. See the index. Access to some examples is forbidden.
Also see Matrices, from Oregon State University's Web Study Guide for Vector Calculus.
Determinants are discussed in Linear Algebra WebNotes. See the index.
Also see Determinants, from OSU's Web Study Guide for Vector Calculus.
Conditions for the existence of
(i) a unique solution;
(ii) no solution;
(iii) an infinity of solutions.
See Using Matrices to Solve Systems with Three or More Unknowns, an on-line tutorial accompanying the book Finite Mathematics & Calculus Applied to the Real World, by Stefan Waner and Steven Costenoble of Hofstra University. The three cases for solutions are explicitly discussed at the end of this tutorial. It is also possible to download a pivoting program for the TI-82 and TI-83 calculators.
See Systems of linear equations,
from Linear Algebra WebNotes.
These notes were written by Mark Sapir, of the University of Nebraska.
Functions from linear algebra can be calculated at Java Script Linear Algebra.
Also see Gaussian Elimination, from Oregon State University's Web Study Guide for Vector Calculus. In this method, row operations are carried out on matrices.
For an introduction to the subject that aims at a "visceral understanding rather than a rigorous logical presentation", see Vectors. Explorations are conducted using an interactive program, The Geometer's Sketchpad. A demo version of the program can be downloaded from the site (Mac or Windows). The site also has a Vector Discussion Group attached to it.
The Graphing Vector Calculator is an interactive Java applet that enables the graphing of two-dimensional vectors and the results of simple operations on them.
See Decomposition of Vectors, from the Vectors site.
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Graphed using Mathematica and METRIC packages. |
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See Vector Length and Unit Vectors, both from the Vectors site.
See Dot Products and Projections, from the Oregon State University Math Department's Web Study Guide.
The component formula for the scalar product, together with examples and problems, can be found at The Scalar Product, from S.O.S. Mathematics.
Also see Definition of the Dot Product, What Good Are Dot Products?, and Projection - Theory, all from the Vectors site from The Math Forum.
From Projection - Numbers. Used with permission.
See The Vector Cross Product - A JAVA Interactive Tutorial, from the University of Syracuse. An interactive diagram of a × b = c allows vectors a and b to be lengthened and shortened, and the angle f between them can be adjusted. The result of these changes can be seen in vector c. The plane containing vectors a and b can also be tilted. Recommended.
Also see The Cross Product, from the OSU Math Department's Web Study Guide.
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Vector i is red. The cross product is shown in magenta, perpendicular to the xy-plane. The diagram was made using Mathematica and METRIC packages. |
The Cross Product provides a good illustration of the vector product, making clear the length of the vector formed when two vectors are crossed.
The different forms of the equations of lines and planes.
See Equations of Lines and Planes, from the Oregon State University Math Department's Web Study Guide.
Also see 12.2 Planes, from Geometry Formulas and Facts. This resource is from the Geometry Center at the University of Minnesota.
The Three Dimensional Graphing Applet can be used to graph a plane expressed in Cartesian form. The plane can be rotated, and the viewer can zoom in and out. The applet was written by James Goodenberger, a fifteen year old high school student.
(i) two lines;
(ii) a line with a plane;
(iii) two planes;
(iv) three planes.
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The red plane has equation z = 3. The equation of the green plane is -4y - 3z = 8. The equation of the cyan plane is 4x + y + 2z = 2. The planes meet at the point (-1.75, 3, 3). Created using Mathematica and packages developed by METRIC. |
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