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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 Hessenberg form#23619
  • Definition of homogeneous linear system of equations#23433
  • Definition of how the action of a linear transformation on a basis extends to the whole space#23867
  • Definition of identity linear transformation#24014
  • Definition of identity matrix#23457
  • Definition of ill-conditioned linear system#23446
  • Definition of image (of a point) under a linear transformation#23842
  • Definition of inconsistent linear system#23432
  • Definition of independent subspaces#23972
  • Definition of index of nilpotency#23584
  • Definition of inner/dot product on C^n#23896
  • Definition of inner/dot product on R^n#23895
  • Definition of inner product#24088
  • Definition of inner product space#24089
  • Definition of intersection of subspaces#23966
  • Definition of invariant subspace of a linear transformation.#24052
  • Definition of inverse of a linear transformation#23849
  • Definition of inverse of a matrix#23517
  • Definition of invertible linear transformation#23848
  • Definition of invertible matrix#23518
  • Definition of invertible/nonsingular linear transformation#24021
  • Definition of isomorphic/isomorphism between vector spaces#24001
  • Definition of Jordan form#23620
  • Definition of kernel/null space of linear transformation#24058
  • Definition of kernel of linear transformation#23847
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