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Definition of basis of a vector space (or subspace)
Definition of the standard/natural basis of R^n (or C^n)
The standard/natural basis of R^n (or C^n) is a basis.
Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)
Definition of the standard basis of the polynomials of degree at most n
Definition of the standard basis of the m by n matrices
Definition of coordinates relative to a given basis
A set of nonzero vectors contains (as a subset) a basis for its span.
The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.
Each vector can be written uniquely as a linear combination of vectors from a given basis.
A set is a basis if each vector can be written uniquely as a linear combination.
Coordinates

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Definition of the standard basis of the m by n matrices

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The set of m by n matrices is a vector space.Close match
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Definition of the standard matrix for a linear transformationClose match
Definition of m by n matrixClose match
Definition of the standard/natural basis of R^n (or C^n)Close match
Basic terminology and notationClose match
Matrices as linear transformationsClose match
Definition of matrixClose match
Definition of matrix multiplicationClose match
Definition of matrix representation of a linear transformation with respect to bases of the spacesClose match
BasesClose match
Notation for entry of matrixClose match
The standard/natural basis of R^n (or C^n) is a basis.Close match
Definition of size of a matrixClose match
Definition of the standard basis of the polynomials of degree at most nClose match
Definition of basis of a vector space (or subspace)Close match
A linear transformation is given by a matrix whose columns are the images of the standard basis vectorsClose match
An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).Close match
CoordinatesClose match
Each vector can be written uniquely as a linear combination of vectors from a given basis.Close match

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