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## List of problems that can be solved using Linear Algebra Decoded

The examples are solutions to problems solved by Linear Algebra Decoded

Matrices, determinants and linear equations | |||

Compute the sum of two matrices. | View example | ||

Compute the difference of two matrices. | |||

Compute the product of a matrix by a scalar. | |||

Compute the product of two matrices. | View example | ||

Compute the product of a square matrix by itself. | |||

Find the transpose of a matrix. | |||

Calculate the determinant of a square matrix. | View example | ||

Transform a matrix to row echelon form using elementary row transformations. | View example | ||

Calculate the rank of a matrix, transforming it first to row echelon form. | |||

Compute the matrix of cofactors. | |||

Compute the adjugate matrix. | |||

Compute the inverse of a matrix using the adjugate matrix. | |||

Compute the inverse of a matrix using row operations. | View example | ||

Find the LU factorization for a matrix. | |||

Classify a system of linear equations. | |||

Solve a system of linear equations using Gaussian elimination. | View example | ||

Vector spaces and subspaces | |||

Express a vector as a linear combination of a set of vectors. | View example | ||

Determine if a set of vectors from a vector space is linearly dependent or independent. | |||

Find the vector subspace spanned by a set of vectors. | View example | ||

Calculate the dimension of a vector subspace expressed by its implicit equations. | |||

Extract a basis from a spanning set. | |||

Find a basis for a vector subspace expressed by its implicit equations. | |||

Determine if a set of vectors is a basis for a subspace expressed by its implicit equations. | |||

Determine if a set of vectors is a basis for the subspace spanned by another set of vectors. | |||

Expand a set of vectors into a basis for the vector space. | |||

Find the coordinate vector of a given vector, relative to a basis for the vector space. | |||

Determine which vectors of a basis for the vector space can be replaced for a given vector, in order that the new set of vectors continues being a basis for the vector space. | |||

Compute the change of basis matrix. | View example | ||

Find the subspace obtained from the intersection of two subspaces which are expressed by their implicit equations. | |||

Find the subspace obtained from the intersection of two subspaces which are given by spanning sets. | |||

Find the subspace obtained from the intersection of two subspaces, where the first one is expressed by its implicit equations, and the second one by a spanning set. | |||

Find the subspace obtained from the sum of two subspaces which are expressed by their implicit equations. | View example | ||

Find the subspace obtained from the sum of two subspaces which are given by spanning sets. | |||

Find the subspace obtained from the sum of two subspaces, where the first one is expressed by its implicit equations, and the second one by a spanning set. | |||

Determine if two subspaces which are expressed by their implicit equations, are complementary subspaces. | |||

Determine if two subspaces which are given by spanning sets, are complementary subspaces. | |||

Determine if two subspaces are complementary subspaces, where the first one is expressed by its implicit equations, and the second one by a spanning set. | |||

Find a complementary subspace for a given subspace expressed by its implicit equations. | |||

Find a complementary subspace for the subspace spanned by a set of vectors. | |||

Determine if two subspaces which are expressed by their implicit equations, are equal. | |||

Determine if two subspaces which are given by spanning sets, are equal. | View example | ||

Determine if two subspaces are equal, where the first one is expressed by its implicit equations, and the second one by a spanning set. | |||

Linear transformations | |||

Find the matrix of a linear transformation with respect to the standard bases. | |||

Find the matrix of a linear transformation with respect to two given bases, one for the input space and the other one for the output space. | |||

Compute the image of a given vector under a linear transformation. | |||

Find a basis and the parametric representation of the kernel (null-space) of a linear transformation. | View example | ||

Find a basis and the implicit equations of the image (range) of a linear transformation. | |||

Determine if the kernel and the image of an endomorphism are complementary subspaces. | |||

Classify a linear transformation. | |||

Find a linear transformation that maps an ordered set of vectors of the input space in an ordered set of vectors of the output space. | View example | ||

Find a linear transformation which image can be spanned by a given set of vectors. | |||

Find the linear transformation resulting from adding two linear transformations. | |||

Find the linear transformation resulting from multiplying a linear transformation by a scalar. | |||

Find the linear transformation resulting from composing two linear transformations. | |||

Find the inverse of a linear transformation. | |||

Find the characteristic polynomial associated to a linear transformation. | |||

Find the eigenvalues and eigenvectors of a linear transformation. | |||

Determine if a linear transformation is diagonalizable. | View example | ||

Determine if a subspace expressed by its implicit equations, is invariant with respect to a linear transformation. | |||

Determine if the subspace spanned by a set of vectors is invariant with respect to a linear transformation. |