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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
  • Two vectors are dependent iff at least one 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 is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.

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526–550 of 718 nodes
  • Orthogonality and projection#23893
  • Parametric form of the solution set of a system of linear equations#23419
  • Parametric vector form of the solution set of a system of linear equations#23420
  • Particular types of matrices#23535
  • Projection#23937
  • Proof of several equivalences for nonsingular matrix#23644
  • Pythagorean equality characterizes orthogonality over the reals#23918
  • QR decomposition#23630
  • Rank and nullity#23882
  • Rank and nullity#23672
  • Rank–nullity theorem for an m-by-n matrix#23679
  • Real block form and complex diagonalization for a 2-by-2 matrix#23618
  • 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 augmented matrices represent equivalent systems#23769
  • Row equivalent matrices have the same row space.#23881
  • 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
  • Scalar multiplication of matrices is associative and distributive#23473
  • Similarity of matrices#23634
  • Similarity of matrices is an equivalence relation#23636
  • Similar matrices have the same eigenvalues and the same characteristic polynomials.#23638
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Container for Linear Algebra#24123
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