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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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  • Definition of distance between vectors#24091
  • Definition of domain of a linear transformation#23840
  • Definition of echelon form of a linear system#23422
  • Definition of (echelon matrix/matrix in (row) echelon form)#23539
  • Definition of eigenspace of a linear transformation#24050
  • Definition of eigenspace of a matrix#23684
  • Definition of eigenvalue/characteristic value of a linear transformation#24042
  • Definition of eigenvalue of a matrix#23681
  • Definition of eigenvector/characteristic vector of a linear transformation#24043
  • Definition of eigenvector of a matrix#23682
  • Definition of elementary matrix#23557
  • Definition of entry/component of a vector#23783
  • Definition of equality of matrices#23459
  • Definition of equality of vectors#23785
  • Definition of equation operations on a linear system#23424
  • Definition of equivalent matrices#23604
  • Definition of equivalent systems of linear equations#23426
  • Definition of extended reduced row echelon form of a matrix#23550
  • Definition of free/independent variable in a linear system#23443
  • Definition of generalized inverse of a matrix#23534
  • Definition of generalized kernel/null space of linear transformation#24060
  • Definition of generalized range space of a linear transformation#24039
  • Definition of geometric multiplicity of an eigenvalue#23697
  • Definition of Gram-Schmidt process#24093
  • Definition of Hermitian/self-adjoint matrix#23597
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