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Linear algebra
Reading time: 2minThis section provides a robust set of functions for working with linear algebra concepts, including matrices, vectors, and their properties. You can define and manipulate matrices, calculate their inverses, determinants, and transposes, and explore essential concepts like eigenvalues and dimensions. These tools are invaluable for creating problems in systems of equations, transformations, and other areas of linear algebra.
Example classroom uses
- Generate random matrices and ask students to find their inverse or transpose, setting logical conditions to check the correctness of their operations.
- Isolate specific columns of a matrix for analysis or manipulation in a problem.
- Construct and use diagonal matrices in problems involving transformations or scaling.
- Present a matrix and ask students to find its eigenvalues, or verify if a given value is an eigenvalue.
- Create problems where students need to determine the dimensions of a matrix or vector, or ensure that matrix operations are performed on compatible dimensions.
👉Need concrete examples? Check out the Examples section to see how these functions are used in real exercises. It’s a great place to see how to combine random value generation, conditions, and solution logic in context.
Functions
adjoint_matrix
Given a matrix, computes its adjoint matrix.
adjoint_matrix(Matrix)constant_matrix
Given two integers n and m and an expression e, constructs a matrix of dimensions n×m with all its entries equal to e.
constant_matrix(Integer, Expression)
constant_matrix(Integer, Integer, Expression)constant_vector
Given an integer n and an expression e, constructs a vector of dimensions n with all its entries equal to e.
constant_vector(Integer, Expression)determinant
Given a matrix A, computes its determinant.
determinant(Matrix)diagonal_matrix
Given a list, range or vector L, constructs a diagonal matrix with the elements of L in the diagonal.
Given an integer n and an expression e, constructs a diagonal matrix of dimension n×n with coefficients equal to e.
diagonal_matrix(List | Range | Vector)
diagonal_matrix(Integer, Expression)dimensions
Given a matrix, returns two numbers: the number of rows and the number of columns.
dimensions(Matrix)eigenvalues
Given a matrix, this function returns a list with its eigenvalues over ℝ (repeated eigenvalues are not omitted).
eigenvalues(Matrix)eigenvectors
Given a matrix, this function returns a matrix with its eigenvectors over ℝ disposed in columns.
eigenvectors(Matrix)identity_matrix
Given an integer n, constructs the identity matrix of dimensions n×n.
identity_matrix(Integer)inverse
Given a matrix, returns its inverse.
inverse(Matrix)jordan
Given a matrix, returns the matrix J. If the option transformation_matrix is set to true, the output is a list with J and P.
jordan(Matrix)
jordan(Matrix, transformation_matrix=trueThe Jordan decomposition of a matrix A is a change of basis where A is written into a diagonal or quasi-diagonal form: A=P-1JP, where P is the change of basis matrix and J is a matrix with the following structure
where λ1,...,λn are the eigenvalues of A.
linearly_independents?
Given a set of vectors v1,...,vn, returns true if they are linearly independents, and false otherwise.
linearly_independents?(Vector, ..., Vector)
linearly_independents?(List)norm
Given a vector, returns its norm.
norm(Vector)n_columns
Given a matrix, returns the number of columns.
n_columns(Matrix)n_rows
Given a matrix, returns the number of rows.
n_rows(Matrix)rank
Given a matrix, returns its rank.
rank(Matrix)trace
Given a matrix, returns its trace.
trace(Matrix)transpose
Given a matrix, returns its transpose.
transpose(Matrix)vectorial_product
Given two vectors in 
, returns its vectorial product.
vectorial_product(Vector, Vector)versor
Given a vector, returns its versor or normalized vector.
versor(Vector)