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  • Linear systems and echelon matrices

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  • The echelon form can be used to determine if a linear system is consistent.
  • Example of using the echelon form to determine if a linear system is consistent.
  • The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.
  • The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.
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  1. The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.

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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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