MTH2021 · Linear Algebra with Applications
Fundamental Subspaces, Fields & Coding Theory
Week 6 attaches four subspaces to every matrix — the row space, column space, kernel (null space) and cokernel — and reads their dimensions off the rank. It also generalises scalars to a field (including Z_p) and applies the ideas to error-correcting codes, notably the Hamming (7,4) code with its generator and parity-check matrices. The four-subspace computations sit in Quiz 6 and Test 2 (Weeks 3–7), and the Fundamental Theorem of Linear Algebra returns in Chapter 10.
What this chapter covers
- 01The four fundamental subspaces: row(A), col(A), ker(A) = {x : Ax = 0}, coker(A) = ker(Aᵀ)
- 02Rank r = number of pivots = dim(row(A)) = dim(col(A)); ker and coker are subspaces
- 03Dimensions: dim row = dim col = r; dim ker(A) = n − r; dim coker(A) = m − r
- 04Ax = b is consistent ⇔ b ∈ col(A)
- 05Fields F (axioms F1–F6); examples ℚ, ℝ, ℂ, Z_p for prime p; vector spaces over a field, Fⁿ
- 06Modular arithmetic in Z_p and binary vector spaces over Z₂
- 07Coding theory: Hamming (7,4) code, generator matrix G (c = dG) and parity-check matrix H (Hcᵀ = 0)
- 08Error detection/correction: a valid codeword satisfies Hcᵀ = 0; a nonzero syndrome locates a single-bit error
The four fundamental subspaces of a 2×3 matrix
- +1Row-reduce: R2 → R2 − 2R1 gives (0, 0, 1), so RREF(A) = [[1, 2, 0], [0, 0, 1]]. There are 2 pivots (columns 1 and 3), so rank r = 2.
- +1Row space: the nonzero rows of the RREF span it. Basis of row(A) = {(1, 2, 0), (0, 0, 1)}, dim = r = 2.
- +1Column space: the pivots are in columns 1 and 3, so take those columns of the ORIGINAL A. Basis of col(A) = {(1, 2)ᵀ, (1, 3)ᵀ}, dim = r = 2 — so col(A) = ℝ².
- +1Kernel: solve Ax = 0. From the RREF, x₁ + 2x₂ = 0 and x₃ = 0 with x₂ free. Set x₂ = 1: x = (−2, 1, 0). Basis of ker(A) = {(−2, 1, 0)}, dim = n − r = 3 − 2 = 1.
- +1Cokernel: coker(A) = ker(Aᵀ) has dim m − r = 2 − 2 = 0, so coker(A) = {0} (basis is the empty set). Dimension check: r + (n−r) = 2 + 1 = 3 = n ✓ and r + (m−r) = 2 + 0 = 2 = m ✓.
Key terms
- Row space row(A)
- The subspace of ℝⁿ spanned by the rows of A; its dimension is the rank r, and the nonzero rows of the RREF form a basis.
- Column space col(A)
- The subspace of ℝᵐ spanned by the columns of A; dim = r, with the original columns in pivot positions forming a basis. Ax = b is solvable iff b ∈ col(A).
- Kernel (null space) ker(A)
- The solution set {x : Ax = 0}, a subspace of ℝⁿ of dimension n − r; its basis comes from the free variables.
- Cokernel coker(A)
- ker(Aᵀ), the left null space, a subspace of ℝᵐ of dimension m − r.
- Field
- A set with addition and multiplication satisfying the field axioms F1–F6 (including inverses), such as ℚ, ℝ, ℂ or Z_p for prime p; scalars for a vector space are drawn from a field.
- Hamming (7,4) code
- An error-correcting binary code that encodes 4 data bits into 7 via a generator matrix G (c = dG); the parity-check matrix H detects and locates a single-bit error through the syndrome Hcᵀ.
Fundamental Subspaces, Fields & Coding Theory FAQ
What are the four fundamental subspaces and how do I find them?
They are the row space and kernel (living in ℝⁿ) and the column space and cokernel (living in ℝᵐ). Row-reduce A once: the pivots give the rank r; nonzero RREF rows span row(A); the original columns in pivot positions span col(A); free variables give ker(A); and coker(A) = ker(Aᵀ) has dimension m − r.
Why do I take columns of the original matrix, not the RREF, for col(A)?
Row operations preserve the row space but change the column space, so the RREF's columns are not columns of A. What is preserved is which columns are pivot columns, so you identify the pivot positions from the RREF and then read those columns off the original A to get a basis for col(A).
How do the four dimensions relate?
dim row(A) = dim col(A) = r, dim ker(A) = n − r and dim coker(A) = m − r. So the domain splits as r + (n − r) = n and the codomain as r + (m − r) = m. These identities are the Fundamental Theorem of Linear Algebra and give you a free self-check on every computation.
Why does the unit bother with fields and Z_p?
Because the whole theory — spans, independence, rank — works over any field, not just the real numbers. Working over Z₂ (bits) turns the machinery into coding theory: the Hamming (7,4) code uses a generator matrix to encode and a parity-check matrix whose syndrome pinpoints a single-bit error, a genuine application of null spaces.
Where is this examined?
The four-subspace computations and modular/Z_p arithmetic are drilled in Quiz 6 and fall inside Test 2 (Weeks 3–7). The coding-theory application and the rank identities can appear in the final, and the row/kernel orthogonality returns as the Fundamental Theorem of Linear Algebra in the Gram–Schmidt chapter.
Exam move
Anchor everything on one row reduction: pivots give rank, nonzero RREF rows give row(A), original pivot columns give col(A), free variables give ker(A), and coker(A) = ker(Aᵀ). Always verify dim row = dim col = r, dim ker = n − r, dim coker = m − r as a check. For the coding-theory application, practise encoding with G and computing the syndrome Hcᵀ over Z₂. This is examined in Quiz 6 and Test 2 (Weeks 3–7). When a subspace basis looks wrong, ask Sia to re-derive it from your RREF.
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