(1) create the matrices you want to use (by going to the MATRIX EDIT option and selecting CREATE NEW), or
(2) load previously saved matrices from disk (by going to the DISK OPS option and selecting LOAD MATRICES).
Add two matrices.
Subtract one matrix from another.
Multiply a matrix by a constant.
Multiply two matrices.
Transposes a matrix.
Calculate the inverse of a square matrix, if it exists.
Raises a square matrix to a positive integer power.
Computes the determinant of a square matrix.
End this madness!
Select this option to do various vector operations.
Before doing this you should either:
(1) create the vectors you want to use (by going to the VECTOR EDIT option and selecting CREATE NEW), or
(2) load previously saved vectors from disk (by going to the DISK OPS option and selecting LOAD VECTORS).
Add two vectors.
Subtract one vector from another.
Multiply a vector by a constant.
Calculates the inner product (dot product, scalar product) of two n-dimensional vectors, and also their lengths, the distance between them, and the angle between them.
Calculates the cross product (vector product) of two 3-dimensional vectors, and determines its length. Recall that the cross product is orthogonal to the original vectors, and is 0 if and only if they are dependent. The length of the cross product is the area of the parallelogram determined by the vectors.
Determines whether a set of m vectors in n-space is linearly dependent or independent.
If the set is linearly dependent, an explicit dependence relation is displayed.
Performs the Gram-Schmidt orthogonalization procedure on m vectors in n-space, producing an orthogonal (or orthonormal) basis for the subspace they span.
This will allow you to save existing matrices and vectors to disk.
It also allows you to load previously saved matrices and vectors from disk, to change drives, and to change directories.
Load a file of matrices from disk. The file should have been created with this program, and will have a filename extension .MAT.
This process will NOT overwrite existing matrices and vectors. If there is insufficient room for all the matrices, it will load as many as it can.
If a loaded matrix has the same name as an existing matrix or vector, you will be asked to rename the loaded matrix.
Load a file of vectors from disk. The file should have been created with this program, and will have a filename extension .VEC.
This process will NOT overwrite existing matrices and vectors. If there is insufficient room for all the vectors, it will load as many as it can.
If a loaded vector has the same name as an existing vector or matrix, you will be asked to rename the loaded vector.
Saves the existing matrices to disk. You will be asked to supply a filename, but will not be allowed to choose the name of an existing file. The file will have a filename extension .MAT.
There is no danger of overwriting an existing file, and this process will NOT delete existing matrices and vectors.
Saves the existing vectors to disk. You will be asked to supply a filename, but will not be allowed to choose the name of an existing file. The file will have a filename extension .VEC.
There is no danger of overwriting an existing file, and this process will NOT delete existing matrices and vectors.
Saves the existing vectors to disk. You will be asked to supply the filename, but you will not be allowed to choose the name of an existing file. The file will have a filename extension .VEC.
This process will NOT delete existing matrices and vectors.
Allows you to select a different disk drive.
Gives information about the current disk drive selected, including total disk space and total available space.
Shows the filenames in the current directory.
Allows you to move to a different directory on the current disk drive.
This allows you to create new matrices. The maximum size of a matrix is 10 by 10. This restriction allows each matrix to be nicely formatted on the screen.
This allows you to edit an existing matrix.
This allows you to look at existing matrices.
This allows you to delete existing matrices.
Allows you to change the name of the matrix.
Allows you to add a remark to the name of a matrix.
This allows you to print matrices.
This allows you to construct augmented matrices from existing matrices and vectors.
This allows you to create new vectors. The maximum dimension of a vector is 10. This restriction allows each vector to be viewed on the screen in a well formatted way.
This allows you to edit an existing vector.
This allows you to look at existing vectors.
This allows you to delete existing vectors.
This allows you to change the name of a vector.
This allows you to add a remark to the name of a vector.
This allows you to print vectors.
This allows you to select the maximum number of decimal places displayed on the screen. The current value is shown on the top right-hand side of the screen. This number is purely for cosmetic purposes. It has NO effect on the accuracy of the calculations. Initially it is set to 5.
This allows you to set the "tolerance" of the calculation. It is initially set to .000000001, which means that all numbers between this and its negative are treated as though they were actually zero. This value DOES affect the numerical accuracy of the results.
This allows you to select one of three options related to having the results of a calculation sent to a printer. They are:
Always print the results (YES),
Never print the results (NO),
Ask whether to print the results (ASK).
The present status of this option (YES, NO, ASK) is always displayed on the top right-hand side of the screen alongside the word PRINT. Initially the print option is set to ASK.
This allows you to select one of three options related to keeping the matrices and vectors which result from a calcu-
lation. The options are:
Always keep the results (YES),
Never keep the results (NO),
Ask whether to keep the results (ASK).
The present status of this option (YES, NO, ASK) is always displayed on the top right-hand side of the screen alongside the word KEEP. Initially the option is set to ASK.
One of the main purposes of this linear algebra package is to encourage students to discover and conjecture results for themselves, in other words, an exploratory tool. To this end we have included a series of projects which the student can be assigned at appropriate times. The projects can be accessed via this option.
This is a table of decimals with their approximate fractional counterparts. For example:
0.1428571429
1/7.
This list can also be accessed when viewing a matrix, by pressing the F10 key. When this is possible an appropriate reminder appears at the bottom of the screen.
This is an ever growing list of students and faculty who have helped with this project.
This allows you to create, edit, print, and view matrices.
This allows you to create, edit, print, and view vectors.
This allows you to perform miscellaneous operations.
This allows you to do elementary row operations on a step-by-step basis. It is always possible to retrace your steps right back to the initial matrix, if desired. Should you choose to have the result printed, only the latest path is followed, i.e. all retracing of steps is ignored.
Enter all the entries. Any entry left blank will automatically be set equal to zero.
You can move around in this matrix/vector by using the arrow keys on the right hand side of the keyboard. Also, the HOME key moves one entry towards the upper left, the END key moves one entry towards the lower right. PgUp takes you directly to the upper left entry, PgDn to the lower right. When you have finished editing, press the ESCAPE key.
Enter the diagonal and upper triangular entries. The lower triangular entries will be entered automatically.
Enter the upper triangular entries. The diagonal and the lower triangular entries will be entered automatically.
Enter the diagonal entries. The off-diagonal entries will automatically be set equal to zero.
Enter the diagonal and upper triangular entries. The remaining entries will automatically be set equal to 0.
Enter the lower triangular entries. The upper triangular entries will be set equal to zero.
Enter the number of rows and columns (between 1 and 10) for the matrix and whether you wish to create it by typing the entries across or down.
This is a typical help screen.
Don't forget to press the ESCape key to remove each help screen.
There are two options for solving systems of linear equations, called SOLVE and LEAST SQUARES.
The current value of the highlighted entry is displayed at the bottom of the screen along with its row and column. You can move around in this matrix/vector by using the arrow keys on the right hand side of the keyboard. The HOME key moves one entry toward the upper left, the END key moves one entry toward the lower right. PgUp takes you directly to the upper left entry, PgDn to the lower right. When you have finished viewing, press the ESCAPE key.
IDENTITY creates a matrix with 1's on the diagonal and 0's elsewhere.
DIAGONAL creates a matrix with 0's off the diagonal. You supply the diagonal entries.
SYMMETRIC creates a matrix which is symmetric about the diagonal. You supply the upper triangular entries.
ANTI-SYMMETRIC, also called SKEW-SYMMETRIC, creates a matrix which has 0's on the diagonal, and the lower triangular entries are the negatives of the upper triangular entries.
<Press PgDn for more>
UPPER TRIANGULAR creates a matrix with zeros in the lower triangular entries. You supply the upper triangular entries.
LOWER TRIANGULAR creates a matrix with zeros in the upper triangular entries. You supply the lower triangular entries.
STOCHASTIC creates a matrix with random non-
negative entries (between 0.0000 and 0.9999), where the sum of the entries in each row is 1.
NONE OF THE ABOVE permits you to select all the matrix entries.
<Press PgUp to go back>
If you answer YES, the entries of the matrix will be randomly chosen between -9.99 and 9.99 . If the matrix is square, you will still be able to select the type of matrix (e.g. symmetric, etc.).
If you answer NO, you must enter the appropriate entries.
Enter the name by which you want to identify this matrix/vector. The name must be unique and less than 6 characters long. You can change the name later by going to the MATRIX EDIT (or VECTOR EDIT) option and selecting CHANGE NAME.
Select a dimension between 2 and 10.
If you answer YES, the entries of the vector will be chosen randomly between -9.99 and 9.99.
If you answer NO, you must enter the appropriate entries.
Make your choice by using the cursor up and down keys, then press the RETURN or ENTER key.
Two options are available, called CHARACTERISTIC POLYNOMIAL and THE POWER METHOD.
The options are ROW OPERATIONS, REDUCED ECHELON, and SIMILARITY TRANSFORMATION.
Calculates the row-reduced echelon form of an m by n matrix.
Calculates the similarity transformation
-1
B AB
of a square matrix A by an invertible matrix B.
Takes you directly to the MATRIX EDITOR.
The "normal equations" method is used to give a best approximate solution to a system of linear equations. The routine is intended for overdetermined (hence inconsistent) systems, but it works for all systems. No provision is made for the best solution having minimal norm in case the best solution is not unique.
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Applies Gauss-Jordan elimination to solve a system AX = B of linear equations, if solutions exist. If multiple solutions exist then a particular solution X = Y is exhibited, and a basis for the null space of A, i.e. for the solution space of the homogeneous system AX = 0. All solutions to AX = B are then of the form Y + Z, where Z is any linear combination of the basis vectors.
Calculates the coefficients of the characteristic polynomial f(x) = det(A - xI) of an n by n matrix A, and exhibits them for decreasing powers of x.
If n = 2 then eigenvalues and eigenvectors are also calculated and displayed.
This allows you to read an extended version of the original instructions.
This allows you to load the previously saved settings for KEEP, PRINT, and DECIMALS.
This allows you to save the existing settings for KEEP, PRINT, and DECIMALS.
Assigned homework can be accessed through this option.
(.txt)