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- 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
- 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 real or complex linear system has zero, one, or infinitely many solutions#23439
- 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 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
- 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.#23824
- A square real matrix has orthonormal columns iff its inverse is its transpose#23930
- A subset of a linearly independent set is linearly independent.#23980
- A vector can be written uniquely as a linear combination of vectors from independent subspaces.#23973
- A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.#23934
- A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.#23923