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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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551–575 of 715 nodes
  • The 0 vector is unique.#23958
  • The 0 vector multiplied by any scalar equals the 0 vector.#23961
  • The additive inverse of a vector equals the vector multiplied by -1.#23963
  • The additive inverse of a vector is called the negative of the vector.#23956
  • The additive inverse of a vector is unique.#23959
  • The adjoint of a matrix-scalar product is the product of the adjoint and the conjugate.#23508
  • The adjoint of a product of matrices is the product of the adjoints in reverse order.#23513
  • The adjoint of a sum is the sum of the adjoints.#23507
  • The Cauchy-Schwartz inequality#23913
  • The Cauchy-Schwarz inequality#24092
  • The Cayley-Hamilton theorem for a linear transformation#24046
  • The Cayley-Hamilton theorem for a matrix.#23707
  • The change of coordinates matrix between two bases exists and is unique#23996
  • The characteristic polynomial applied to the matrix gives the 0 matrix.#23706
  • The column space of a matrix is a vector space#24109
  • The column space of an m-by-n matrix is a subspace of R^m (or C^m)#23876
  • The composition of injective linear transformations is injective#24025
  • The composition of invertible linear transformations is invertible#24027
  • The composition of linear transformations is a linear transformation#24024
  • The composition of surjective linear transformations is surjective#24026
  • The condition number of matrix measures how close it is to being singular#23750
  • The conjugate of a matrix-scalar product is the product of the conjugates.#23468
  • The conjugate of a product of matrices is the product of the conjugates.#23469
  • The conjugate of a sum of vectors in C^n is the sum of the conjugates#23791
  • The conjugate of the sum of matrices is the sum of the conjugates.#23467
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