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Linear transformations

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Eigenvalues and eigenvectors
The matrix representation of a sum of linear transformations is the sum of the matrices.
A linear transformation is invertible if and only if it is injective and surjective
Subspaces associated to a linear transformation
A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
The preimage of a vector is a translation of the kernel of the linear transformation
The matrix representation of a composition of linear transformations is the product of the matrices.
Geometric properties of linear transformations
Matrices as linear transformations
Basic properties of linear transformations
Description of a spanning set for the null space of a matrix from the reduced row-echelon form.
Description of a basis for the null space of a matrix from the reduced row-echelon form.
The nonzero rows of an echelon form of a matrix are linearly independent.
Subspaces associated to a matrix
Rank and nullity
Examples
Terminology
Composition
The image of a linearly independent set under an injective linear transformation is linearly independent.
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.
The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.
A linear transformation is surjective if and only if the rank equals the dimension of the codomain.
The range of a linear transformation is a subspace
The image of a spanning set spans the range
A linear transformation is surjective if and only if the image of a basis is a spanning set
Definition of generalized range space of a linear transformation
A linear transformation is injective on its generalized range space.
Definition of diagonalizable linear transformation
The rank plus the nullity of a linear transformation equals the dimension of the domain.
Linear maps preserve dependence of lists, with image-set caveats
A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
Definition of matrix representation of a linear transformation with respect to bases of the spaces
A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces
Definition of matrix representation of a linear transformation from a vector space to itself
The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.
The matrix of an inverse is the inverse of the matrix representation
Eigenvalues agree and eigenvectors correspond through coordinates
Triangularization requires the characteristic polynomial to split
Equivalence theorems for injective transformations

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Matrix describing a rotation of the plane
Example of a linear transformation on R^2: generic
Example of a linear transformation on R^2: shear
Example of a linear transformation on R^2: rotation
Example of a linear transformation on R^2: projection
Example of a linear transformation on R^3: rotation

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Matrices as linear transformationsClose match
A linear transformation is given by a matrix whose columns are the images of the standard basis vectorsClose match
Example of a linear transformation on R^3: rotationClose match
Example of a linear transformation on R^2: genericClose match
Geometric properties of linear transformationsClose match
Geometric properties of linear transformations on R^2Close match
Example of a linear transformation on R^2: rotationClose match
Example of a linear transformation on R^2: projectionClose match
Example of a linear transformation on R^2: shearClose match
Matrices act as a transformation by multiplying vectorsClose match
Visualise a linear transformation on R^2 by looking at the image of a regionClose match
A linear transformation is given by multiplying by its matrix representation with respect to bases of the spacesClose match
Basic properties of linear transformationsClose match
CompositionClose match
A linear transformation is determined by its action on a basis.Close match
Matrix describing a rotation of the planeClose match
Definition of matrix representation of a linear transformation from a vector space to itselfRelated
ProjectionRelated
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The composition of linear transformations is a linear transformationRelated

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