Linear algebra extension for Numbas

This extension provides a few functions which make working with linear algebra easier.

The three kinds of permitted row operation are:

• Swap two rows.
• Multiply a row by a scalar.
• Subtract k times one row from another.

This extension works with matrices over the rationals: before any operations are performed, the matrix's cells are converted to fractions.

JME functions

row_echelon_form(matrix)

Uses row operations to put the given matrix in row echelon form. A matrix is in row echelon form if the leading non-zero entry in each row is strictly to the right of the leading non-zero entries in all of the rows above.

row_echelon_form_display(matrix)

Returns a passage of HTML describing the steps involved in transforming the given matrix into row echelon form.

row_echelon_form_display_determinant(matrix)

Returns a passage of HTML describing the steps involved in transforming the given matrix into row echelon form, while describing how the determinant of the matrix changes at each step.

is_row_echelon_form(matrix)

Returns true if the matrix is in row echelon form.

describe_why_row_echelon_form(matrix)

Returns a string describing why the matrix is not in row echelon form, or "The matrix is in row echelon form." if it is.

reduced_row_echelon_form(matrix)

Uses row operations to put the given matrix in reduced row echelon form. A matrix in row echelon form is reduced if every leading non-zero entry is 1 and is the only non-zero entry in its column.

reduced_row_echelon_form_display(matrix)

Returns a passage of HTML describing the steps involved in transforming the given matrix into reduced row echelon form.

is_reduced_row_echelon_form(matrix)

Returns true if the matrix is in reduced row echelon form.

describe_why_reduced_row_echelon_form(matrix)

Returns a string describing why the matrix is not in reduced row echelon form, or "The matrix is in reduced row echelon form." if it is.

rank(matrix)

Computes the rank of the matrix, by returning the number of rows of the reduced row echelon form of the matrix, with all zero rows removed.

is_linearly_independent(matrix)

Returns true if the rows of the matrix are linearly independent.

adjoin(matrix,vector)

Add the given column vector to the right of the given matrix.

subset_with_dimension(vectors,n,d)

Return a subset of n of the given vectors, with dimension d.

This is not always possible - if the vectors have length k, you can't have d>k. This is not If the input list has dimension less than d, it can't be done. This is not Likewise with extra dependent vectors - if there aren't enough, it'll fail.

The vectors are processed in order, so if you want a random subset you should shuffle the list first.

as_sum_of_basis(basis,v)

Express the vector v as the sum of the given list of basis vectors. Returns a list of coefficients corresponding to the basis vectors.

JavaScript functions

All JavaScript functions are members of the Numbas.extensions.linearalgebra object.

Fraction(n)

A constructor for a fraction. n is either a number, in which case a rational approximation is computed, or an object {n: numerator, d:denominator}.

Example:

var two = new Fraction(2); var third = new Fraction(1/3); var two_thirds = two.mul(third);

fraction_matrix(matrix)

Convert all the entries in the given matrix to Fraction objects. Returns a new matrix - doesn't modify the original.

unfraction_matrix(matrix)

Convert all the entries in the given matrix from Fraction objects to normal numbers. Returns a new matrix - doesn't modify the original.

row_echelon_form(matrix)

Transforms the given matrix into row echelon form. Returns an object {matrix: matrix, operations: [list of descriptions of the steps]}.

reduced_row_echelon_form(matrix)

Transforms the given matrix into reduced row echelon form. Returns an object {matrix: matrix, operations: [list of descriptions of the steps]}.

is_row_echelon_form(matrix)

Returns true if the given matrix is in row echelon form. If not, throws an error whose message is an explanation of why the matrix isn't in row echelon form.

is_reduced_row_echelon_form(matrix)

Returns true if the given matrix is in reduced row echelon form. If not, throws an error whose message is an explanation of why the matrix isn't in reduced row echelon form.

rank(matrix)

Returns the rank of the given matrix.

is_linearly_independent(vectors)

Returns true if the given list of vectors is linearly independent.

adjoin(matrix,vector)

Add the given column vector to the right of the given matrix.

subset_with_dimension(vectors,n,d)

Returns a subset of n of the given vectors with dimension d.

as_sum_of_basis(basis,v)

Express the vector v as the sum of the given basis vectors. Returns a list of coefficients corresponding to the basis vectors.