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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. Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors

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  • Example of vector-scalar multiplication in R^2#23794
  • Example of writing a given vector in R^3 as a linear combination of given vectors#23803
  • Examples#23886
  • Examples of vector spaces#24095
  • Factorization of matrices#23622
  • Finite-dimensional Fitting decomposition into generalized kernel and range#24061
  • Finite-dimensional vector spaces of equal dimension over the same field are isomorphic#24005
  • F^n is a vector space.#24099
  • Formula for computing the least squares solution to a linear system#23949
  • Formula for computing the least squares solution to a linear system.#23948
  • Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.#23933
  • Formula for the coordinates of the projection of a vector onto a subspace#23944
  • Formula for the determinant of a 2-by-2 matrix.#23742
  • Formula for the determinant of a 3-by-3 matrix.#23743
  • Formula for the inverse of a 2-by-2 matrix.#23521
  • Formula for the least-squares linear fit to 2-dimensional data#23952
  • Formula for the (orthogonal) projection of one vector onto another vector#23939
  • Formula for the spectral decomposition for a symmetric matrix#23576
  • For n-by-n invertible matrices A and B#23523
  • Gaussian elimination as a method to solve a linear system#23421
  • Gauss-Jordan procedure to put a matrix into reduced row echelon form#23540
  • Geometric description of span of a set of vectors in R^n (or C^n)#23806
  • Geometric picture of a 2-by-2 linear system#23428
  • Geometric picture of a 3-by-3 linear system#23430
  • Geometric picture of the solution set of a linear equation in 3 unknowns#23429
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Container for Linear Algebra#24123
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  • Test Flight Problem Set#23151

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