Kunze Linear Algebra · Problem 2.2.5
Vector Spaces
Mathematics — Methods · subspaces-of-matrix-spaces
Problem
Let be a field and let be a positive integer (). Let be the vector space of all matrices over . Which of the following sets of matrices in are subspaces of ? (a) all invertible ; (b) all non-invertible ; (c) all such that , where is some fixed matrix in ; (d) all such that .Intuitive explanation
To identify subspaces, remember that a subset must be closed under addition and scalar multiplication, and contain the zero vector. For each set, ask: does it include the zero matrix? If you add two matrices from the set, do you stay in the set? If you scale a matrix by a scalar, do you stay in the set? The key insight is that invertibility and idempotence are fragile properties that break under these operations, while commutativity with a fixed matrix is robust—it's preserved by linear combinations.
Intuitive explanation
To identify subspaces, remember that a subset must be closed under addition and scalar multiplication, and contain the zero vector. For each set, ask: does it include the zero matrix? If you add two matrices from the set, do you stay in the set? If you scale a matrix by a scalar, do you stay in the set? The key insight is that invertibility and idempotence are fragile properties that break under these operations, while commutativity with a fixed matrix is robust—it's preserved by linear combinations.