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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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  • Abstract vector spaces#23953
  • A consistent system with more variables than equations has infinitely many solutions.#23440
  • Adding a multiple of one row to another row does not change the determinant.#23737
  • Addition#23461
  • A determinant is a multilinear function#23736
  • A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns.#23615
  • Adjoining an element not in the span of a linearly independent set gives another linearly independent set.#23983
  • A homogeneous system has a nontrivial solution if and only if it has a free variable.#23437
  • A homogeneous system with more variables than equations has infinitely many solutions.#23441
  • Algebraic properties of R^n (or C^n)#23777
  • Algorithm for computing an LU decomposition#23626
  • A linear system is equivalent to a matrix equation.#23753
  • A linear system is equivalent to a vector equation.#23752
  • A linear transformation has a representation as an upper triangular matrix.#24076
  • A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.#24075
  • A linear transformation is determined by its action on a basis.#23866
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.#24054
  • A linear transformation is given by a matrix whose columns are the images of the standard basis vectors#23858
  • A linear transformation is given by a matrix with respect to a given basis.#23859
  • A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces#24069
  • A linear transformation is injective on its generalized range space.#24040
  • A linear transformation is invertible if and only if it is injective and surjective#24067
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.#24066
  • A linear transformation is surjective if and only if the columns of its matrix span the codomain.#23885
  • A linear transformation is surjective if and only if the image of a basis is a spanning set#24038
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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