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  • Linear systems and echelon matrices

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  • The echelon form can be used to determine if a linear system is consistent.
  • Example of using the echelon form to determine if a linear system is consistent.
  • The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.
  • The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.
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  1. The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.

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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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