Abstract vector spaces
A consistent system with more variables than equations has infinitely many solutions.
Adding a multiple of one row to another row does not change the determinant.
Addition
A determinant is a multilinear function
A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns.
Adjoining an element not in the span of a linearly independent set gives another linearly independent set.
A homogeneous system has a nontrivial solution if and only if it has a free variable.
A homogeneous system with more variables than equations has infinitely many solutions.
Algebraic properties of R^n (or C^n)
Algorithm for computing an LU decomposition
A linear system is equivalent to a matrix equation.
A linear system is equivalent to a vector equation.
A linear transformation has a representation as an upper triangular matrix.
A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.
A linear transformation is determined by its action on a basis.
A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
A linear transformation is given by a matrix whose columns are the images of the standard basis vectors
A linear transformation is given by a matrix with respect to a given basis.
A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces
A linear transformation is injective on its generalized range space.
A linear transformation is invertible if and only if it is injective and surjective
A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
A linear transformation is surjective if and only if the columns of its matrix span the codomain.
A linear transformation is surjective if and only if the image of a basis is a spanning set

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