Source: Friedberg, Insel & Spence, Linear Algebra 4th Ed., Ch. 1.3
Tags: subspace, subspace test, closed under addition, closed under scalar multiplication, transpose, symmetric matrix, diagonal matrix, trace, intersection of subspaces, direct sum
Difficulty: Foundational Prerequisites: Vector space definition and axioms (Sections 1.1–1.2 study notes). Familiarity with matrices and polynomials.
Once you know what a vector space is, the natural next question is: which subsets of a vector space are themselves vector spaces? These are called subspaces, and they appear everywhere in linear algebra. The critical insight is that you do not need to check all eight axioms from scratch. Six of them are inherited automatically from the parent space, so the subspace test reduces to just three conditions. This section also introduces the transpose, symmetric matrices, diagonal matrices, and the trace, all of which are tools you will use repeatedly.
A subspace of a vector space V is a subset W that is itself a vector space under the same operations. The practical test (Theorem 1.3) requires only three checks: W contains the zero vector, W is closed under addition, and W is closed under scalar multiplication. Intersections of subspaces are always subspaces; unions generally are not.
Subspace
A subset W of a vector space V over a field F that is a vector space over F using the same addition and scalar multiplication defined on V.
In simple terms, a smaller "world" sitting inside a vector space where you can still add and scale freely without leaving that world.
Closed under addition
A subset W is closed under addition if, whenever x and y are in W, their sum x + y is also in W.
Closed under scalar multiplication
A subset W is closed under scalar multiplication if, whenever c is in F and x is in W, the product cx is also in W.
Zero subspace
The subspace {0}, containing only the zero vector. It is a subspace of every vector space.
Transpose (A^t)
The n × m matrix obtained from an m × n matrix A by swapping rows and columns: (A^t){ij} = A{ji}.
In simple terms, flip the matrix along its main diagonal.
Symmetric matrix
A square matrix A satisfying A^t = A. Equivalently, A_{ij} = A_{ji} for all i, j.
Think of it as: the matrix is a mirror image of itself across the diagonal.
Skew-symmetric matrix
A square matrix M satisfying M^t = −M. All diagonal entries must be zero (in fields where 1 + 1 ≠ 0).
Diagonal matrix
A square matrix where all entries off the main diagonal are zero: M_{ij} = 0 whenever i ≠ j.
Upper triangular matrix
An m × n matrix where all entries below the diagonal are zero: A_{ij} = 0 whenever i > j.
Trace (tr(M))
The sum of the diagonal entries of a square matrix: tr(M) = M₁₁ + M₂₂ + ... + Mₙₙ.
In simple terms, add up the numbers running from top-left to bottom-right.
A subset W of a vector space V is a subspace of V if and only if:
(a) 0 ∈ W (the zero vector of V is in W)
(b) x + y ∈ W whenever x, y ∈ W (closed under addition)
(c) cx ∈ W whenever c ∈ F and x ∈ W (closed under scalar multiplication)
Why only three conditions? The remaining axioms (VS 1, VS 2, VS 5, VS 6, VS 7, VS 8) hold for all vectors in V, so they automatically hold for vectors in any subset. Condition (c) with c = −1 gives −x = (−1)x ∈ W, so additive inverses come for free. And condition (a) ensures the zero vector is the right one (the same as V's zero vector, by the cancellation law).
W ≠ ∅ and, whenever a ∈ F and x, y ∈ W, then ax ∈ W and x + y ∈ W (Exercise 17)
0 ∈ W and ax + y ∈ W whenever a ∈ F and x, y ∈ W (Exercise 18, a single combined closure condition)
Polynomials of bounded degree:
P_n(F), the set of polynomials in P(F) of degree ≤ n, is a subspace of P(F).
Zero polynomial has degree −1 ≤ n ✓
Sum of polynomials of degree ≤ n has degree ≤ n ✓
Scalar times a polynomial of degree ≤ n has degree ≤ n ✓
Continuous functions:
C(R), the set of all continuous real-valued functions on R, is a subspace of F(R, R).
The zero function is continuous ✓
Sum of two continuous functions is continuous ✓
Scalar times a continuous function is continuous ✓
Diagonal matrices:
The set of n × n diagonal matrices is a subspace of M_{n×n}(F).
Off-diagonal entries of A + B: 0 + 0 = 0 ✓
Off-diagonal entries of cA: c·0 = 0 ✓
Symmetric matrices:
The set W of symmetric matrices in M_{n×n}(F) is a subspace.
O^t = O, so O ∈ W ✓
(A + B)^t = A^t + B^t = A + B when both are symmetric ✓
(cA)^t = cA^t = cA when A is symmetric ✓
Matrices with trace zero:
The set of n × n matrices with tr(M) = 0 is a subspace of M_{n×n}(F).
Uses the linearity of trace: tr(aA + bB) = a·tr(A) + b·tr(B)
The set of matrices in M_{m×n}(R) with all nonnegative entries is not a subspace, because multiplying by a negative scalar produces negative entries. It fails closure under scalar multiplication.
(A + B)^t = A^t + B^t
(cA)^t = cA^t
(aA + bB)^t = aA^t + bB^t
(A^t)^t = A
A + A^t is always symmetric
tr(aA + bB) = a·tr(A) + b·tr(B) (trace is linear)
Diagonal matrices are symmetric
Theorem 1.4: Any intersection of subspaces of V is a subspace of V.
Proof idea: 0 is in every subspace, so it is in the intersection. Closure under addition and scalar multiplication follows because each subspace in the collection is closed.
Union warning: The union of two subspaces W₁ ∪ W₂ is a subspace if and only if one contains the other (W₁ ⊆ W₂ or W₂ ⊆ W₁). In general, unions are not subspaces because they may fail closure under addition.
Sum: If S₁, S₂ are nonempty subsets of V, then S₁ + S₂ = {x + y : x ∈ S₁, y ∈ S₂}.
Direct sum: V = W₁ ⊕ W₂ means W₁ and W₂ are subspaces with W₁ ∩ W₂ = {0} and W₁ + W₂ = V.
W₁ + W₂ is always a subspace containing both W₁ and W₂ (Exercise 23).
W₁ + W₂ is the smallest subspace containing both.
V is the direct sum of W₁ and W₂ if and only if every vector in V can be uniquely written as x₁ + x₂ with x₁ ∈ W₁ and x₂ ∈ W₂ (Exercise 30).
Transpose:
(A^t){ij} = A{ji}
Trace:
tr(M) = M₁₁ + M₂₂ + ... + Mₙₙ
Linearity of trace:
tr(aA + bB) = a·tr(A) + b·tr(B)
Subspaces capture the idea of "constraints within a system." In structural engineering, the set of all equilibrium force configurations on a truss forms a subspace of the full vector space of possible force assignments. In data science, the set of all images that can be described by a given set of principal components is a subspace of "all possible images."
Students often forget to check that 0 ∈ W. If the defining condition does not hold at the zero vector (e.g., W = {(a₁,a₂,a₃) : a₁ + 2a₂ − 3a₃ = 1}), the set is not a subspace, because (0,0,0) does not satisfy the equation.
A subset that is a vector space under different operations from the parent space is not a subspace. The operations must be inherited, not redefined.
The union of two subspaces is almost never a subspace. Students sometimes assume it is because intersection works.
The trace is the sum of diagonal entries, not the product. This is a common slip in exam conditions.
⚠️ The subspace test (Theorem 1.3) is the workhorse of this section. Expect to apply it to specific sets defined by equations (like {(a₁,a₂,a₃) : 2a₁ − 7a₂ + a₃ = 0}).
⚠️ Sets defined by homogeneous linear equations (= 0) are subspaces. Sets defined by nonhomogeneous equations (= c, c ≠ 0) are not, because 0 is not in the set.
⚠️ You will be asked to prove that specific matrix sets (symmetric, diagonal, upper triangular, trace-zero) are subspaces. Know the transpose properties to make these proofs clean.
⚠️ The intersection theorem (Thm 1.4) and the union characterisation (Exercise 19) are common exam questions.
⚠️ Direct sums appear in later chapters. Understand the definition and the unique-decomposition characterisation.
True or false: the empty set is a subspace of every vector space.
True or false: if V is a vector space other than {0}, then V contains a subspace W with W ≠ V and W ≠ {0}.
Fill in the blank: the trace of a square matrix is the ______ of its diagonal entries.
True or false: the intersection of any two subsets of V is a subspace of V.
True or false: an n × n diagonal matrix can have more than n nonzero entries.
Answers: 1. False (∅ has no zero vector). 2. True (for any nonzero vector v, span({v}) works if dim V > 1; if dim V = 1, then {0} and V are the only subspaces, but V ≠ {0} is given). 3. Sum. 4. False (they must be subspaces, not just subsets). 5. False (only the n diagonal positions can be nonzero).
Q: Determine whether W = {(a₁, a₂, a₃) ∈ R³ : a₁ + 2a₂ − 3a₃ = 1} is a subspace of R³.
A: No. The zero vector (0, 0, 0) does not satisfy 0 + 0 − 0 = 1, so 0 ∉ W. Hence W is not a subspace.
Q: Determine whether W = {(a₁, a₂, a₃) ∈ R³ : 2a₁ − 7a₂ + a₃ = 0} is a subspace of R³.
A: Yes. (0,0,0) satisfies 0 = 0. For closure: if 2a₁ − 7a₂ + a₃ = 0 and 2b₁ − 7b₂ + b₃ = 0, then 2(a₁+b₁) − 7(a₂+b₂) + (a₃+b₃) = 0. And 2(ca₁) − 7(ca₂) + (ca₃) = c(2a₁ − 7a₂ + a₃) = 0. All three conditions of Theorem 1.3 hold.
Q: Prove that the set of upper triangular n × n matrices is a subspace of M_{n×n}(F).
A: The zero matrix has all entries zero, so A_{ij} = 0 for i > j. If A and B are upper triangular, then for i > j, (A+B){ij} = A{ij} + B_{ij} = 0 + 0 = 0. For scalar multiplication, (cA)_{ij} = c·0 = 0 for i > j. By Theorem 1.3, it is a subspace.
Q: Prove that the union of two subspaces W₁ and W₂ is a subspace only if one contains the other.
A: If neither W₁ ⊆ W₂ nor W₂ ⊆ W₁, pick x ∈ W₁ \ W₂ and y ∈ W₂ \ W₁. If W₁ ∪ W₂ were a subspace, x + y ∈ W₁ ∪ W₂. If x + y ∈ W₁, then y = (x + y) − x ∈ W₁, contradicting y ∉ W₁. Similarly for W₂. Contradiction. Conversely, if W₁ ⊆ W₂, then W₁ ∪ W₂ = W₂, which is a subspace.
Subspaces connect directly to span (Section 1.4): the span of any subset S is always a subspace, and in fact it is the smallest subspace containing S. The concept also feeds into dimension theory (Section 1.6), where the dimension of a subspace is always at most the dimension of the parent space. In later chapters, the null space and range of a linear transformation are defined as subspaces, making this test essential for Chapters 2 and beyond.
subspace, subspace test, Theorem 1.3, closed under addition, closed under scalar multiplication, zero subspace, transpose, symmetric matrix, skew-symmetric matrix, diagonal matrix, upper triangular matrix, trace, intersection of subspaces, direct sum, coset, quotient space, Friedberg linear algebra chapter 1