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Lesson 111 — Vector spaces: axioms, subspaces, basis, dimension

Abstract definition via 8 axioms. Subspaces, linear combination, basis, dimension. The leap from arrow geometry to the algebraic structure that supports all modern linear algebra.

Used in: Leistungskurs Algebra Linear (Klasse 12 alemã) · H2 Mathematics — Singapura · Math III japonês (vetores e espaços)

\dim V = n \iff V \cong K^n

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Rigorous notation, full derivation, hypotheses

Axiomatic definition

The 8 vector space axioms

Definition· K-vector space

Let $K$ be a field (typically $\mathbb{R}$ or $\mathbb{C}$ ). A $K$ -vector space is a nonempty set $V$ equipped with two operations:

addition: $+ : V \times V \to V$
scalar multiplication: $\cdot : K \times V \to V$

satisfying, for all $u, v, w \in V$ and $a, b \in K$ :

#	Axiom	Formula
1	Associativity	$(u + v) + w = u + (v + w)$
2	Commutativity	$u + v = v + u$
3	Additive identity	$\exists\, \mathbf{0} \in V : v + \mathbf{0} = v$
4	Additive inverse	$\exists\, (-v) \in V : v + (-v) = \mathbf{0}$
5	Vector distributivity	$a(u + v) = au + av$
6	Scalar distributivity	$(a + b)v = av + bv$
7	Compatibility	$a(bv) = (ab)v$
8	Scalar identity	$1 \cdot v = v$

"We will use the term field to refer to either $\mathbb{R}$ or $\mathbb{C}$ . Elements of a field are called scalars. A vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on $V$ such that the following properties hold..." — Axler — Linear Algebra Done Right, §1A

"A subset $W$ of $V$ is called a subspace of $V$ if $W$ is a vector space under the addition and scalar multiplication defined on $V$ ." — Beezer — A First Course in Linear Algebra, §VS

Definition· Linear combination, span, linear independence, basis

Given $S = \{v_1, \ldots, v_k\} \subseteq V$ :

Linear combination: $c_1 v_1 + \cdots + c_k v_k$ with $c_i \in K$ .
Span (generated space): $\text{span}(S) = \{c_1 v_1 + \cdots + c_k v_k : c_i \in K\}$ — the smallest subspace containing $S$ .
Linearly independent (LI): $c_1 v_1 + \cdots + c_k v_k = \mathbf{0} \Rightarrow c_1 = \cdots = c_k = 0$ .
Basis: $\mathcal{B} \subseteq V$ that is LI and $\text{span}(\mathcal{B}) = V$ .
Dimension: $\dim V = |\mathcal{B}|$ — well-defined by the theorem below.

SVG — Concept hierarchy

Hierarchy: field provides scalars; V satisfies 8 axioms; basis has dim V elements; classification gives V ≅ Kⁿ.

Worked examples

Example— 1· Verification of 8 axioms for polynomials (application)

Problem: Show that $\mathcal{P}_2 = \{a_0 + a_1 x + a_2 x^2 : a_i \in \mathbb{R}\}$ with usual polynomial addition and real scalar multiplication is a vector space over $\mathbb{R}$ .

Strategy: Verify the 8 axioms directly for generic polynomials. Most inherit properties from the reals (field).

Solution:

Take $p = a_0 + a_1 x + a_2 x^2$ and $q = b_0 + b_1 x + b_2 x^2$ in $\mathcal{P}_2$ , and $c, d \in \mathbb{R}$ .

Closure: $p + q = (a_0+b_0) + (a_1+b_1)x + (a_2+b_2)x^2 \in \mathcal{P}_2$ . $cp = ca_0 + ca_1 x + ca_2 x^2 \in \mathcal{P}_2$ .

Axiom 1 (assoc.): $(p+q)+r = p+(q+r)$ — coefficients are real, associativity inherited.

Axiom 2 (comut.): $p+q = q+p$ — coefficients are real, commutativity inherited.

Axiom 3 (identity): $\mathbf{0} = 0 + 0x + 0x^2$ . Clearly $p + \mathbf{0} = p$ .

Axiom 4 (inverse): $-p = -a_0 - a_1 x - a_2 x^2 \in \mathcal{P}_2$ and $p + (-p) = \mathbf{0}$ .

Axiom 5: $c(p+q) = c(a_0+b_0) + \ldots = cp + cq$ . Distributivity of reals.

Axiom 6: $(c+d)p = (c+d)a_0 + \ldots = cp + dp$ . Same.

Axiom 7: $c(dp) = c(da_i x^i) = (cd)a_i x^i$ . Associativity of real multiplication.

Axiom 8: $1 \cdot p = 1 \cdot a_0 + 1 \cdot a_1 x + 1 \cdot a_2 x^2 = p$ .

All 8 axioms verified. Thus $\mathcal{P}_2$ is an $\mathbb{R}$ -vector space.

Verification: The canonical basis $\{1, x, x^2\}$ is LI (none is a linear combination of the others) and generates $\mathcal{P}_2$ . Thus $\dim \mathcal{P}_2 = 3$ , consistent with 3 free coefficients.

Source. Beezer — A First Course in Linear Algebra, §VS Example VS.EXS — GNU FDL.

Example— 2· Verify subspace: symmetric matrices (intermediate)

Problem: Show that $W = \{A \in \mathcal{M}_{2 \times 2} : A^T = A\}$ ( $2\times2$ symmetric matrices) is a subspace of $\mathcal{M}_{2\times2}$ .

Strategy: Check the 3 subspace criteria: zero, closure under addition, and closure under scalars.

Solution:

Zero: The null matrix satisfies $\mathbf{0}^T = \mathbf{0}$ , hence $\mathbf{0} \in W$ .

Closure under addition: Let $A, B \in W$ , i.e., $A^T = A$ and $B^T = B$ . Then: $(A + B)^T = A^T + B^T = A + B.$ Thus $A + B \in W$ .

Closure under scalar: Let $c \in \mathbb{R}$ and $A \in W$ . Then: $(cA)^T = c A^T = c A.$ Thus $cA \in W$ .

All 3 criteria verified: $W$ is a subspace.

Verification: A basis of $W$ is $\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix}, \begin{pmatrix}0&1\\1&0\end{pmatrix}, \begin{pmatrix}0&0\\0&1\end{pmatrix}\right\}$ — three LI matrices, hence $\dim W = 3$ . Makes sense: symmetry imposes $a_{12} = a_{21}$ , leaving 3 free entries ( $a_{11}, a_{12}, a_{22}$ ).

Source. Hefferon — Linear Algebra, Chapter 2 §2.I Exercise 2.I.1.17 — CC-BY-SA.

Example— 3· Find basis and dimension of a span (intermediate)

Problem: Find a basis and dimension of the subspace $W = \text{span}\{v_1, v_2, v_3\}$ where $v_1 = (1, 2, 3)$ , $v_2 = (2, 4, 6)$ , $v_3 = (1, 0, 1)$ .

Strategy: Form matrix with vectors as rows and row-reduce. Nonzero rows of the reduced form are a basis of the span.

Solution:

$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 0 & 1 \end{pmatrix} \xrightarrow{L_2 - 2L_1} \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{pmatrix} \xrightarrow{L_3 - L_1} \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & -2 & -2 \end{pmatrix} \xrightarrow{\text{swap } L_2, L_3} \begin{pmatrix} 1 & 2 & 3 \\ 0 & -2 & -2 \\ 0 & 0 & 0 \end{pmatrix}$

Two nonzero rows. A basis of $W$ :

$\mathcal{B} = \{(1, 2, 3),\ (0, -2, -2)\} \quad \text{or equivalently} \quad \{(1, 2, 3),\ (0, 1, 1)\}$

Thus $\dim W = 2$ .

Verification: $v_2 = 2v_1$ (dependent — confirms it doesn't contribute). $v_3 = v_1 - (0,2,2)/2 \cdot 1 + \ldots$ — can be written as a combination of $(1,2,3)$ and $(0,1,1)$ . The span is a plane in $\mathbb{R}^3$ .

Source. Beezer — A First Course in Linear Algebra, §LI Example LI.LIHS — GNU FDL.

Example— 4· Coordinates in a noncanonical basis (advanced)

Problem: Let $\mathcal{B} = \{b_1, b_2\} = \{(1, 1), (1, -1)\}$ be a basis of $\mathbb{R}^2$ . Find the coordinates of vector $v = (5, 1)$ in this basis.

Strategy: Solve the system $c_1 b_1 + c_2 b_2 = v$ for the scalars $c_1, c_2$ — the coordinates of $v$ in basis $\mathcal{B}$ .

Solution:

$c_1 \begin{pmatrix}1\\1\end{pmatrix} + c_2 \begin{pmatrix}1\\-1\end{pmatrix} = \begin{pmatrix}5\\1\end{pmatrix}$

System: $\begin{cases} c_1 + c_2 = 5 \\ c_1 - c_2 = 1 \end{cases}$

Adding: $2c_1 = 6 \Rightarrow c_1 = 3$ . Subtracting: $2c_2 = 4 \Rightarrow c_2 = 2$ .

The coordinates of $v$ in basis $\mathcal{B}$ are $[v]_\mathcal{B} = (3, 2)$ .

Verification: $3(1,1) + 2(1,-1) = (3,3) + (2,-2) = (5, 1)$ . Correct.

Note: The vector $v = (5,1)$ is the same geometric object; what changes are its numeric representations depending on the chosen basis. Change of basis only changes the coordinates, not the vector.

Source. Axler — Linear Algebra Done Right, §2B — CC-BY-NC.

Example— 5· Proof: 0 cdot v = 0 using the axioms (proof)

Problem: Using only the 8 vector space axioms, prove that $0 \cdot v = \mathbf{0}$ for all $v \in V$ (where $0$ is the scalar zero and $\mathbf{0}$ is the additive identity).

Strategy: The scalar $0$ does not appear directly in the axioms. The strategy is to use $0 = 0 + 0$ (zero property in the scalars) and axiom 6 (scalar distributivity), then cancel using the inverse.

Solution:

Let $w = 0 \cdot v$ . We want to show $w = \mathbf{0}$ .

Step 1. In field $K$ , we have $0 = 0 + 0$ (zero is additive identity in $K$ ).

Step 2. By axiom 6 (scalar distributivity): $0 \cdot v = (0 + 0) \cdot v = 0 \cdot v + 0 \cdot v = w + w.$

Step 3. Thus $w = w + w$ .

Step 4. Adding $(-w)$ (inverse of $w$ , axiom 4) to both sides: $w + (-w) = (w + w) + (-w) = w + (w + (-w)) \quad \text{(axiom 1)}$ $\mathbf{0} = w + \mathbf{0} \quad \text{(axiom 4 and 3)}$ $\mathbf{0} = w.$

Therefore $0 \cdot v = \mathbf{0}$ . $\blacksquare$

Verification: The result is a theorem, not an axiom — it follows from the 8. The proof uses: $K$ is a ring, axioms 1 (assoc.), 3 (identity), 4 (inverse), 6 (scalar distrib.). No axiom was used unnecessarily.

Source. Axler — Linear Algebra Done Right, §1A, Proposition 1.2 — CC-BY-NC.

Exercise list

40 exercises · 10 with worked solution (25%)

Application 21Understanding 6Modeling 7Challenge 2Proof 4

Ex. 111.1Application
Verify that $\mathbb{R}^2$ with componentwise addition and scalar multiplication is a vector space over $\mathbb{R}$ .
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Ex. 111.2Application
Verify that $W = \{(x, y) \in \mathbb{R}^2 : x + y = 0\}$ is a subspace of $\mathbb{R}^2$ .
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Ex. 111.3Application
Is the set $W = \{(x, y) \in \mathbb{R}^2 : x + y = 1\}$ a subspace of $\mathbb{R}^2$ ?
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Ex. 111.4Application
Show that $W = \{(x, y, z) \in \mathbb{R}^3 : x = y\}$ is a subspace of $\mathbb{R}^3$ . What is its dimension?
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Ex. 111.5ApplicationAnswer key
Is the set $\{(x, y) \in \mathbb{R}^2 : x \geq 0\}$ (right half-plane) a subspace of $\mathbb{R}^2$ ?
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Ex. 111.6ApplicationAnswer key
Do polynomials of exactly degree 3 form a subspace of $\mathcal{P}_3$ ?
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Ex. 111.7Application
Do even polynomials (with $p(-x) = p(x)$ ) form a subspace of $\mathcal{P}_4$ ? If yes, find a basis.
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Ex. 111.8Application
Show that the $n \times n$ symmetric matrices form a subspace of $\mathcal{M}_{n \times n}$ . For $n = 3$ , what is the dimension?
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Ex. 111.9Application
Are the $n \times n$ invertible matrices a subspace of $\mathcal{M}_{n \times n}$ ?
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Ex. 111.10Application
Show that the solution set of $Ax = \mathbf{0}$ (the kernel of $A$ ) is a subspace of $\mathbb{R}^n$ .
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Ex. 111.11Application
If $V$ and $W$ are subspaces of the same space, is $V \cap W$ a subspace? What about $V \cup W$ ?
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Ex. 111.12Understanding
Show that $W = \{f \in C[0,1] : f(0) = 0 \text{ and } f(1) = 0\}$ is a subspace of $C[0,1]$ (continuous functions on $[0,1]$ ).
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Ex. 111.13Understanding
Is the set of solutions of $Ax = b$ (with $b \neq \mathbf{0}$ ) a subspace of $\mathbb{R}^n$ ?
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Ex. 111.14Proof
Prove that $a \cdot \mathbf{0} = \mathbf{0}$ for every scalar $a \in K$ using only the 8 axioms.
Ex. 111.15Application
Write $(3, 4)$ as a linear combination of $(1, 0)$ and $(0, 1)$ .
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Ex. 111.16Application
Write $(5, 7)$ as a linear combination of $(1, 1)$ and $(1, -1)$ .
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Ex. 111.17Application
Is the set $\{(1, 2),\ (2, 4)\}$ linearly independent in $\mathbb{R}^2$ ?
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Ex. 111.18ApplicationAnswer key
Check whether $\{(1, 2, 3),\ (4, 5, 6),\ (7, 8, 9)\}$ is linearly independent in $\mathbb{R}^3$ .
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Ex. 111.19Application
Show that $\{1, x, x^2\}$ is a basis of $\mathcal{P}_2$ . What is the dimension of $\mathcal{P}_2$ ?
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Ex. 111.20Application
Determine whether $\{(1,1,0),\ (1,0,1),\ (0,1,1)\}$ is a basis of $\mathbb{R}^3$ .
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Ex. 111.21ApplicationAnswer key
Find the coordinates of $(2, 3, 5)$ in the basis $\{(1,1,0),\ (1,0,1),\ (0,1,1)\}$ of $\mathbb{R}^3$ .
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Ex. 111.22ApplicationAnswer key
What is the dimension of the space spanned by $\{(1,0,0),\ (0,1,0),\ (1,1,0),\ (0,0,1)\}$ in $\mathbb{R}^3$ ?
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Ex. 111.23Application
What is the dimension of the space of symmetric $3 \times 3$ matrices?
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Ex. 111.24Application
What is the dimension of $\mathcal{P}_5$ (polynomials of degree $\leq 5$ )?
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Ex. 111.25Understanding
Solutions of $y'' + y = 0$ form a space of what dimension? Exhibit a basis.
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Ex. 111.26UnderstandingAnswer key
Show that if $\dim V = n$ (finite), then every proper subspace of $V$ has dimension strictly less than $n$ .
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Ex. 111.27Modeling
Word2Vec word embeddings live in $\mathbb{R}^{300}$ . What is the dimension of this space? Why 300 and not the vocabulary size?
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Ex. 111.28ModelingAnswer key
Markowitz portfolio with 5 assets lives in $\mathbb{R}^5$ . What dimension has the subspace of portfolios with $\sum w_i = 1$ ?
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Ex. 111.29Modeling
Robot state: position $(x, y)$ , orientation $\theta$ , velocities $(\dot{x}, \dot{y}, \dot{\theta})$ . What is the dimension of the state space?
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Ex. 111.30Modeling
Audio signal at 44,100 Hz, 10 s. What is the dimension of the space without compression? How does MP3 compression exploit effective dimension?
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Ex. 111.31ModelingAnswer key
MNIST image $28 \times 28$ lives in $\mathbb{R}^{784}$ . PCA reveals ~100 relevant components. What does this say about the "effective dimension" of the digit-image subspace?
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Ex. 111.32Modeling
In linear control, $x \in \mathbb{R}^n$ and $u \in \mathbb{R}^m$ . What is the dimension of the $(x, u)$ pair space? What is the controllable subspace?
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Ex. 111.33Modeling
In quantitative finance, replicable payoffs with $n$ assets in $N$ scenarios form a subspace of $\mathbb{R}^N$ . What does "complete market" mean in terms of dimension?
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Ex. 111.34Understanding
What is the dimension of the space of $n \times n$ antisymmetric matrices (with $A^T = -A$ )? For $n = 3$ , exhibit a basis.
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Ex. 111.35Understanding
State and prove (sketch) the Grassmann formula: $\dim(U + W) = \dim U + \dim W - \dim(U \cap W)$ .
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Ex. 111.36Proof
Using only the 8 axioms, prove that $(-1) \cdot v = -v$ for all $v \in V$ .
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Ex. 111.37ProofAnswer key
Prove: if $S$ is linearly dependent, there exists $v_j \in S$ such that $\text{span}(S) = \text{span}(S \setminus \{v_j\})$ .
Ex. 111.38Proof
Prove: if $\dim V = n$ and $W$ is a proper subspace ( $W \subsetneq V$ ), then $\dim W \leq n-1$ .
Ex. 111.39ChallengeAnswer key
Exhibit an explicit basis of $\mathcal{M}_{2 \times 2}$ ( $2\times2$ matrices) and calculate the dimension.
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Ex. 111.40Challenge
Argue that $\mathcal{F}(\mathbb{R}, \mathbb{R})$ (all real functions) has infinite dimension. Use functions with point support.
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Sources

A First Course in Linear Algebra — Robert A. Beezer · 2022 · EN · GNU FDL · §VS, §S, §LC, §LI, §B, §D. Primary source of exercises.
Linear Algebra — Jim Hefferon · 2020 · EN · CC-BY-SA · Chapter 2 (Vector Spaces), §2.I–2.III. Source of examples and chapter 2 exercises.
Linear Algebra Done Right — Sheldon Axler · 2024 · 4th ed · EN · CC-BY-NC · Chapters 1–2. Source of proofs and formal door.