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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.
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The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.

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The pivot columns of a matrix are a basis for the column space.Close match
Description of a basis for the null space of a matrix from the reduced row-echelon form.Close match
Nonzero rows of reduced row-echelon form give a basis for the row spaceClose match
Example of putting a matrix in echelon form and identifying the pivot columnsClose match
Description of a spanning set for the null space of a matrix from the reduced row-echelon form.Close match
Row operations do not necessarily preserve the column space.Close match
A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.Close match
Definition of (row) echelon form of a matrixClose match
A set of nonzero vectors contains (as a subset) a basis for its span.Close match
Definition of pivot columnClose match
BasesClose match
Definition of basis of a vector space (or subspace)Close match
Each vector can be written uniquely as a linear combination of vectors from a given basis.Close match
Echelon matricesClose match
A set is a basis if each vector can be written uniquely as a linear combination.Close match
A linear transformation is given by a matrix with respect to a given basis.Close match
Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrixClose match
The union of bases from independent subspaces is a basis for the space.Close match
The nonzero rows of an echelon form of a matrix are linearly independent.Close match
Definition of the standard basis of the m by n matricesClose match

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