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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 homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.

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  • Definition of leading entry in a row of a matrix#23481
  • Definition of least-squares error of a linear system#23947
  • Definition of least-squares solution to a linear system#23946
  • Definition of left inverse of a matrix#23515
  • Definition of length/norm of a vector#24090
  • Definition of linear combination of vectors#23800
  • Definition of linear dependence relation on a set of vectors#23816
  • Definition of linear equation#23413
  • Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors#23821
  • Definition of linearly independent set of vectors: if a linear combination is 0#23819
  • Definition of linear transformation/homomorphism#24013
  • Definition of LU decomposition#23625
  • Definition of Markov matrix#23595
  • Definition of matrix#23449
  • Definition of matrix diagonalization#23610
  • Definition of matrix equation#23764
  • Definition of matrix in reduced row echelon form#23543
  • Definition of matrix multiplication#23489
  • Definition of matrix multiplication in terms of column vectors#23488
  • Definition of matrix null space (left)#23874
  • Definition of matrix null space (right)#23873
  • Definition of matrix representation of a linear system#23759
  • Definition of matrix representation of a linear transformation#23861
  • Definition of matrix representation of a linear transformation from a vector space to itself#24070
  • Definition of matrix representation of a linear transformation with respect to bases of the spaces#24068
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