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).
An n-by-n matrix nas n (complex) eigenvalues
An n-by-n matrix with n distinct eigenvalues is diagonalizable.
A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations
A nonsingular matrix can be written as a product of elementary matrices.
An orthogonal set of nonzero vectors is linearly independent.
Any linearly independent set can be expanded to a basis for the (sub)space
Any matrix times the 0 matrix equals the 0 matrix.
Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.
Application Leontief input-output analysis
Applications
Applications of band matrices
Applications to cubic spline
Applications to differential equations
Applications to error-correcting code
Applications to Markov chains
Applications to voting and social choice
A scalar multiple of a linear transformation is a linear transformation
A set is a basis if each vector can be written uniquely as a linear combination.
A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.
A set of nonzero vectors contains (as a subset) a basis for its span.
A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.
A set of vectors containing fewer elements than the dimension of the space cannot span
A set of vectors containing more elements than the dimension of the space must be linearly dependent
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.

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