A set of matrices (or functions) defined by a null-space or column-space membership condition forms a vector subspace if and only if it is closed under addition and scalar multiplication, which can be verified directly from the linearity of the defining condition; a set defined by a column-space condition, by contrast, may fail to be a subspace because it excludes the zero vector. This topic belongs to abstract linear algebra's theory of vector spaces, generalizing the notion of subspace beyond Euclidean vectors to spaces whose elements are matrices, and connects to finding a basis and dimension for such subspaces via row-wise linear constraints derived from the null-space condition.
A set of matrices (or functions) defined by a null-space or column-space membership condition forms a vector subspace if and only if it is closed under addition and scalar multiplication, which can b…