The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).
The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).
The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).
The crazy vector space is a vector space.
The determinant function exists.
The determinant function is unique.
The determinant of a block diagonal matrix is the product of the determinants of the blocks.
The determinant of a matrix can be computed as a cofactor expansion across any row.
The determinant of a matrix can be computed as a cofactor expansion down any column.
The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form.
The determinant of a matrix measures the area/volume of the parallelogram/parallelipiped determined by its columns.
The determinant of a triangular matrix is the product of the entries on the diagonal.
The determinant of the inverse of A is the reciprocal of the determinant of A.
The determinant of the matrix of a linear transformation is the factor by which the area/volume changes.
The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.
The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.
The dimension of a subspace is less than or equal to the dimension of the whole space
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.
The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.
The direct sum of a subspace and its orthogonal complement is the whole space.
The echelon form can be used to determine if a linear system is consistent.
The eigenspace of a linear transformation is a nontrivial subspace.
The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation.
The eigenvalues of a polynomial of a matrix are the polynomial of the eigenvalues.
The eigenvalues of a power of a matrix are the power the eigenvalues.

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