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- 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).#23616
- An n-by-n matrix nas n (complex) eigenvalues#23698
- An n-by-n matrix with n distinct eigenvalues is diagonalizable.#23617
- A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations#23965
- A nonsingular matrix can be written as a product of elementary matrices.#23558
- An orthogonal set of nonzero vectors is linearly independent.#23927
- Any linearly independent set can be expanded to a basis for the (sub)space#23984
- Any matrix times the 0 matrix equals the 0 matrix.#23491
- Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.#24061
- Application Leontief input-output analysis#24118
- Applications#24113
- Applications of band matrices#24116
- Applications to cubic spline#24120
- Applications to differential equations#24114
- Applications to error-correcting code#24117
- Applications to Markov chains#24115
- Applications to voting and social choice#24119
- A scalar multiple of a linear transformation is a linear transformation#24018
- A set is a basis if each vector can be written uniquely as a linear combination.#23988
- A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.#23981
- A set of nonzero vectors contains (as a subset) a basis for its span.#23985
- A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.#23826
- A set of vectors containing fewer elements than the dimension of the space cannot span#23979
- A set of vectors containing more elements than the dimension of the space must be linearly dependent#24012
- A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.#23823
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