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  • Linear (in)dependence

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  • Definition of linear dependence relation on a set of vectors
  • Definition of trivial linear dependence relation on a set of vectors
  • Determine if a particular set of vectors in R^3 in linearly independent
  • Definition of linearly independent set of vectors: if a linear combination is 0
  • Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors
  • Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors
  • Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0
  • 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.
  • 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.
  • If a set of vectors contains the 0 vector
  • A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.
  • If a set of vectors in R^n (or C^n) contains more than n elements
  • A subset of a linearly independent set is linearly independent.
  • A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.
  • Removing a linearly dependent vector from a set does not change the span of the set.
  • Adjoining an element not in the span of a linearly independent set gives another linearly independent set.
  • Any linearly independent set can be expanded to a basis for the (sub)space
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  1. 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.

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  • The eigenvalues of a polynomial of a matrix are the polynomial of the eigenvalues.#23691
  • The eigenvalues of a power of a matrix are the power the eigenvalues.#23690
  • The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.#23689
  • The eigenvalues of a triangular matrix are the entries on the main diagonal.#23709
  • The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.#23692
  • The eigenvectors of a normal matrix are an orthonormal basis.#23602
  • The geometry of linear systems#23427
  • The Gram-Schmidt process converts a linearly independent set into an orthogonal set.#23936
  • The identity matrix is the identity for matrix multiplication.#23498
  • The image of a linearly dependent set under a linear transformation is linearly dependent.#24065
  • The image of a linearly independent set under an injective linear transformation is linearly independent.#24032
  • The inner product of a vector with itself is the square of its norm/length.#23910
  • The intersection of subspaces is a subspace#23967
  • The inverse image of a subspace under a linear transformation is a subspace.#24057
  • The inverse of a linear transformation is a linear transformation#24028
  • The inverse of a matrix can be expressed in terms of its matrix of cofactors.#23721
  • The inverse of a matrix can be used to solve a linear system.#23532
  • The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix.#23527
  • The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.#23565
  • The inverse of an isomorphism is an isomorphism.#24002
  • The inverse of a scalar multiple is the reciprocal times the inverse.#23525
  • The inverse of the inverse of a linear transformation is the original linear transformation#24029
  • The kernel/null space of a linear transformation is a subspace#24059
  • The kernels of powers of a linear transformation form an ascending chain#24062
  • The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.#23950
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