The matrix representation of a composition of linear transformations is the product of the matrices.
The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.
The matrix representation of a sum of linear transformations is the sum of the matrices.
The matrix representation of the inverse of linear transformations is the inverse of the matricix.
The minimal polynomial of a linear transformation exists and is unique.
The minimal polynomial of a square matrix exists and is unique.
The nonzero rows of an echelon form of a matrix are linearly independent.
The nonzero rows of the reduced row-echelon form of a matris are a basis for the row space.
The null space of a matrix is a subspace of R^n (or C^n).
The null space of a matrix is the orthogonal complement of the column space.
The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.
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 solutions to a linear system
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
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0
Theorem characterizing when a space is the direct sum of two subspaces
Theorem describing matrix multiplication
Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrix
Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrix
Theorem describing the determinants of elementary matrices.
Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrix
Theorem describing the vector form of sulutions to a linear system.
The orthogonal complement of a subspace is a subspace.
The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector.
The permutation expansion for determinants

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