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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. 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

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526–550 of 715 nodes
  • Proof of several equivalences for nonsingular matrix#23644
  • QR decomposition#23630
  • Rank and mullity#23672
  • Rank and nullity#23882
  • Removing a linearly dependent vector from a set does not change the span of the set.#23982
  • R^n is a vector space.#24097
  • Row equivalence is an equivalence relation#23480
  • Row equivalent matrices have the same row space.#23881
  • Row equivalent matrices represent equivalent linear systems#23769
  • Row operations#23474
  • Row operations are given by multiplication by elementary matrices.#23559
  • Row operations do not necessarily preserve the column space.#23878
  • Scalar multiplication#23470
  • Similarity of matrices#23634
  • Similarity of matrices in an equivalence relation.#23636
  • Similar matrices have the same eigenvalues and the same characteristic polynomials.#23638
  • Spans#23804
  • Subspaces#23811
  • Subspaces associated to a linear transformation#24055
  • Subspaces associated to a matrix#23872
  • Switching two rows multiplies the determinant by -1.#23738
  • Symmetric matrices#23571
  • Symmetric matrices are square.#23573
  • Terminology#23754
  • The 0 scalar multiplied by any vector equals the 0 vector.#23960
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