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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 is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.

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  • A linear transformation is surjective if and only if the rank equals the dimension of the codomain.#24035
  • A linear transformation maps 0 to 0.#23864
  • A linear transformation of a linear combination is the linear combination of the linear transformation#23865
  • A linear transformation on a finite dimentional nontrivial vector space has at least one eigenvalue.#24049
  • All echelon forms of a linear system have the same free variables#23423
  • A matrix and its transpose have the same determinant.#23728
  • A matrix and its transpose have the same eigenvalues/characteristic polynomial.#23693
  • A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose.#23930
  • A matrix equation is equivalent to a linear system#23767
  • A matrix is called ill-conditioned if it is nearly singular#23749
  • A matrix is nilpotent if and only if its only eigenvalue is 0.#23583
  • A matrix is orthogonally diagonalizable if and only if it is normal (The principal axis theorem).#23601
  • A matrix is orthogonally diagonalizable if and only if it is symmetric.#23578
  • A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.#23606
  • A matrix turns into its adjoint when moved to the other side of the standard inner product on C^n.#23906
  • A matrix with a 0 row/column has determinant 0#23741
  • A matrix with real entries and orthonormal columns preserves dot products.#23932
  • A matrix with real entries and orthonormal columns preserves norms.#23931
  • A matrix with real entries has eigenvalues occurring in conjugate pairs.#23710
  • A matrix with two equal rows/columns has determinant 0#23740
  • An eigenspace of a matrix is a nontrivial subspace.#23685
  • An eigenspace of a matrix is the null space of a related matrix.#23686
  • An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.#23612
  • An n-by-n matrix is diagonalizable if and only if the characteristic polynomial factors completely#23614
  • An n-by-n matrix is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n.#23613
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