Module 5: Vector Spaces and Subspaces

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Learning outcomes

After this module, you should be able to:

• verify the vector-space axioms for a given set;
• test whether a subset is a subspace;
• determine linear independence and find a basis;
• define dimension and compute coordinates relative to a basis;
• describe a linear transformation and its kernel/image;
• work with orthogonal vectors, orthogonal complements, projections, and orthogonal/orthonormal bases.

Prerequisites

Complete Module 4 first. Review its quick-revision section if any term below feels unfamiliar.

Short study blocks

Study one row at a time. Draw the vectors whenever the example is in two or three dimensions.

Block	Topic	Suggested time
1	Vector spaces and subspaces	10-15 minutes
2	Span, linear independence, basis, and dimension	10-15 minutes
3	Linear transformations, kernel, and image	10-15 minutes
4	Orthogonality and projections	10-15 minutes
5	Orthogonal bases and Gram-Schmidt	10-15 minutes

Start here: the simple idea

A vector space is any set where you can add elements and scale them and have that behave like arrows in 2D/3D space. A basis is a minimal set of "building blocks" you need to reach every element. Orthogonality means "perpendicular," which lets you project cleanly and solve least-squares problems.

Everyday analogy

• Vector space = a Lego construction mat: any position on the mat is reachable by combining basic step vectors.
• Basis = the smallest set of Lego pieces from which you can build everything.
• Orthogonality = two walls at right angles: moving along one doesn't affect the other, so you can project a point straight onto each wall.

Vector spaces

Definition and axioms

A vector space V over ℝ (or ℂ) is a set with addition and scalar multiplication satisfying 8 axioms (closure + associativity + commutativity + identity + inverses for addition; and compatible scalar distributive laws + 1·v = v). You rarely re-derive all eight; you check closure under + and scalar ×.

Examples:

• ℝⁿ (n-tuples), ℝ² and ℝ³ (the plane, 3D space).
• P_n = polynomials of degree ≤ n (e.g., a₀ + a₁x + a₂x²).
• M_{m×n} = all m×n matrices.
• ℂⁿ = complex n-tuples.

Worked check

Is the set S = { (x, y) ∈ ℝ² : x + y = 0 } a vector space? Check closure:

• Sum: (a,−a) + (b,−b) = (a+b, −(a+b)) ∈ S. ✔
• Scalar: k(a,−a) = (ka, −ka) ∈ S. ✔ So S is a subspace (and thus a vector space) of ℝ².

Subspaces

A subset W of V is a subspace iff:

1. 0 ∈ W (contains the zero vector).
2. Closed under +: u, v ∈ W ⇒ u + v ∈ W.
3. Closed under scalar ×: u ∈ W, c scalar ⇒ c·u ∈ W.

Examples of subspaces of ℝ³: the zero vector, any line through origin, any plane through origin. A line not through the origin is not a subspace (because 0 is not on it).

Span and linear independence

The span of vectors v₁,…,vₖ is all their linear combinations c₁v₁+…+cₖvₖ.

Vectors are linearly independent if the only solution to c₁v₁+…+cₖvₖ = 0 is c₁=…=cₖ=0; otherwise linearly dependent.

Worked example: Are v₁=(1,2) and v₂=(3,4) linearly independent in ℝ²?

c1(1,2) + c2(3,4) = 0  ⇒  c1+3c2=0, 2c1+4c2=0
⇒ c1 = −3c2; 2(−3c2)+4c2 = −2c2 = 0 ⇒ c2=0, c1=0. Independent.

Basis and dimension

A basis of V is a linearly independent set whose span is all of V. The dimension is the number of vectors in any basis.

• ℝⁿ has dimension n; the standard basis is e₁=(1,0,…), e₂=(0,1,0,…), ….
• P₂ has dimension 3; basis {1, x, x²}.
• If you have more vectors than the dimension, they're dependent.

Coordinates

Relative to a basis B = {b₁,…,bₙ}, every v has a unique coordinate vector [v]_B = (c₁,…,cₙ) with v = c₁b₁ + … + cₙbₙ.

Linear transformations

A linear transformation T: V → W satisfies T(u+v) = T(u)+T(v) and T(cu) = cT(u).

• Kernel (null space): {v : T(v) = 0}.
• Image (range): {T(v) : v ∈ V}.
• Matrix representation: once bases are chosen, every linear T is T(v) = Av for a unique matrix A. (This is why matrices are so central.)

Orthogonal vectors and spaces

Vectors u, v are orthogonal iff u·v = 0. The orthogonal complement of a subspace S is S⊥ = {v : v·u = 0 for all u ∈ S}.

• (S⊥)⊥ = S (for subspaces of ℝⁿ).
• Fundamental theorem of linear algebra: row space ⊥ null space, and column space ⊥ left null space, in ℝⁿ and ℝᵐ.

Projections

The orthogonal projection of vector b onto a non-zero vector a is:

proj_a(b) = ( (a·b) / (a·a) ) · a

Geometrically it's the "shadow" of b onto a. If you project onto a subspace (spanned by an orthonormal basis), you sum the projections onto each basis vector.

Worked example

Project b = (3, 1) onto a = (2, 2):

a·b = 6+2 = 8;  a·a = 4+4 = 8
proj = (8/8)·(2,2) = (2,2)

So the foot of the perpendicular from (3,1) onto the line y = x is (2,2).

Orthogonal bases

• An orthogonal basis has mutually perpendicular vectors.
• An orthonormal basis is orthogonal with each vector unit length.
• Advantage: coordinates are just dot products (no solving linear systems).
• Gram-Schmidt process converts any basis of a subspace into an orthogonal basis; normalise to get orthonormal.

Common mistakes

• Thinking any set containing the zero vector is independent (it's dependent — 1·0 = 0).
• Believing a spanning set is always a basis (only if independent).
• Projecting onto a non-unit vector without the a·a divisor.
• Confusing kernel (domain → 0) with image (domain → codomain range).

Memory rules

• Vector space: closed under + and scalar ×; check 0 + closure.
• Subspace: contains 0, closed under +, closed under scalar ×.
• Independent = only trivial combination gives 0; basis = independent + spanning.
• dim(ℝⁿ) = n; standard basis e_i.
• Linear T: T(u+v)=T(u)+T(v), T(cu)=cT(u); represented by a matrix.
• Orthogonal: u·v = 0; proj_a(b) = ((a·b)/(a·a))a.

Mini Quiz

Attempt all questions before revealing the answers.

1. Is the union of two subspaces always a subspace? (yes/no)
2. What is the dimension of P₄ (polynomials of degree ≤ 4)?
3. Are (1,0,0), (0,1,0), (0,0,1) linearly independent?
4. If T is linear and T(u)=0, what is u called?
5. True or false: if u·v = 0 then u and v are orthogonal.

Answers

Reveal answers

1. No (the union need not be closed under addition); the sum is a subspace.
2. 5 (basis {1, x, x², x³, x⁴}).
3. Yes.
4. u is in the kernel (null space) of T.
5. True (by definition of orthogonality).

Quick revision box

• Vector space: closed under + and scalar ×; ℝⁿ, P_n, matrices are examples.
• Subspace: contains 0, closed under + and scalar ·.
• span(S)=all combos; independent = only 0 combo; basis = indep+spanning.
• dim(V)=#basis vectors; ℝⁿ has dim n; coordinates are unique per basis.
• Linear T: T(u+v)=T(u)+T(v), T(cu)=cT(u); kernel={v:T(v)=0}, image=range.
• Orthogonal u·v=0; orthogonal complement S⊥; projection proj_a(b)=((a·b)/(a·a))a.
• Orthogonal/orthonormal bases; Gram-Schmidt orthogonalises a basis.

Exam guidance

Define the notation, state the rule or theorem, show every calculation, and verify or interpret the final result.

Practice ladder

1. Easy - Recall: Define the module's central idea in one or two sentences.
2. Easy - Recognize: Identify the correct method for a small example and explain why it fits.
3. Medium - Apply: Work through one representative problem without copying the example.
4. Medium - Compare: Contrast two methods or concepts from the module.
5. Hard - Integrate: Solve a university-style scenario and justify every major step.

Reveal self-evaluation guide

A complete response uses correct terminology, shows intermediate steps, connects the result to the scenario, and states one assumption or limitation.

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