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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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  • Definition of orthogonal subspaces#24094
  • Definition of orthogonal vectors#23916
  • Definition of orthonormal basis of a (sub)space#23929
  • Definition of orthonormal set of vectors#23926
  • Definition of parallel vectors#23917
  • Definition of permutation matrix#23555
  • Definition of pivot#23476
  • Definition of pivot column#23477
  • Definition of pivot position#23478
  • Definition of positive-definite matrix#23713
  • Definition of pre-image (of a point) under a linear transformation#23843
  • Definition of pre-image of linear transformation#24019
  • Definition of QR decomposition#23631
  • Definition of quadratic form#24122
  • Definition of range of a linear transformation#24020
  • Definition of range of linear transformation#23846
  • Definition of rank factorization of a matrix#23628
  • Definition of rank of a linear transformation#23883
  • Definition of rank of a matrix#23674
  • Definition of rational form#23621
  • Definition of reduced LU decomposition#23627
  • Definition of reduced row echelon form of a matrix#23544
  • Definition of right inverse of a matrix#23516
  • Definition of R^n (or C^n)#23781
  • Definition of (row) echelon form of a matrix#23542
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