The Four Fundamental Subspaces & Rank-Nullity Theorem
“Column space, null space, row space, left null space, orthogonal complements, and dim(V) = rank + nullity.”
1.The Four Fundamental Subspaces of Linear Algebra
§ 1.1.For any real m x n matrix A with rank r:
1. Column Space C(A): Subspace of R^m spanned by the columns of A. dim(C(A)) = r.
2. Null Space N(A): Subspace of R^n containing all solutions to Ax = 0. dim(N(A)) = n - r.
3. Row Space C(A^T): Subspace of R^n spanned by the rows of A. dim(C(A^T)) = r.
4. Left Null Space N(A^T): Subspace of R^m containing all solutions to A^T y = 0. dim(N(A^T)) = m - r.
Fundamental Theorem of Linear Algebra (Gilbert Strang):
N(A) is the orthogonal complement of C(A^T) in R^n.
N(A^T) is the orthogonal complement of C(A) in R^m.
Every vector x in R^n can be uniquely decomposed as x = x_r + x_n, where x_r in C(A^T) and x_n in N(A).
2. Worked Practice Problem: Complete Decomposition of Four Subspaces
Pivot column indices in RREF point back to original matrix columns for C(A).
Problem: For matrix A = [ [1, 2, 0, 1], [0, 1, 1, 0], [1, 3, 1, 1] ] (size 3 x 4), find the rank, nullity, and bases for all four fundamental subspaces.
Step 1: Reduce A to RREF:
R3 -> R3 - R1: [ 0, 1, 1, 0 ]
R3 -> R3 - R2: [ 0, 0, 0, 0 ]
R1 -> R1 - 2*R2: [ 1, 0, -2, 1 ]
RREF(A) = [ [1, 0, -2, 1], [0, 1, 1, 0], [0, 0, 0, 0] ]
Pivot columns: Col 1 and Col 2. Free variables: Col 3 and Col 4.
Rank r = 2, Nullity = n - r = 4 - 2 = 2.
Step 2: Basis for Column Space C(A) (dim = 2 in R^3):
Take original columns 1 and 2 of A:
Basis(C(A)) = { [1, 0, 1]^T, [2, 1, 3]^T }.
Step 3: Basis for Row Space C(A^T) (dim = 2 in R^4):
Take non-zero rows of RREF(A):
Basis(C(A^T)) = { [1, 0, -2, 1]^T, [0, 1, 1, 0]^T }.
Step 4: Basis for Null Space N(A) (dim = 2 in R^4):
x1 = 2*x3 - x4; x2 = -x3; x3 = s; x4 = t.
Basis(N(A)) = { [2, -1, 1, 0]^T, [-1, 0, 0, 1]^T }.
Step 5: Basis for Left Null Space N(A^T) (dim = m - r = 3 - 2 = 1 in R^3):
Solve A^T y = 0: y1 + y3 = 0, 2y1 + y2 + 3y3 = 0, y2 + y3 = 0.
y3 = 1 => y1 = -1, y2 = -1.
Basis(N(A^T)) = { [-1, -1, 1]^T }.
Conclusion: All four dimensions satisfy: dim(C(A))=2, dim(N(A))=2, dim(C(A^T))=2, dim(N(A^T))=1.
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