The pivot columns of a matrix are a basis for the column space.
The preimage of a vector is a translation of the kernel of the linear transformation
The product of square matrices is nonsingular if and only if each factor is nonsingular.
The product of upper/lower triangular matrices is upper/lower triangular.
The projection of a vector which is in a subspace is the vector itself.
The QR decomposition of a nonsingular matrix exists.
The range/image of a linear transformation is a subspace.
The range of a linear transformation is a subspace
The range spaces of powers of a linear transformation form a descending chain
The rank of a matrix equals number of pivots in a reduced row echelon form.
The rank of a matrix equals the rank of the linear transformation it represents.
The rank plus the nullity of a linear transformation equals the dimension of the domain.
The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.
The row space and the column space of a matrix have the same dimension.
The row space of a matrix is a vector space
The set containing only 0 is a vector space.
The set of all functions on a set is a vector space.
The set of all polynomials is a vector space.
The set of all polynomials of degree at most n is a vector space.
The set of all sequences is a vector space.
The set of linear transformations between two vector spaces is a vector space.
The set of m by n matrices is a vector space.
The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.
The solutions to a homogeneous linear differential equation is a vector space.
The solutions to a homogeneous system of linear equations is a vector space.

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