Part I: Foundations — Chapter 1

Vector Spaces

Axioms, subspaces, span, linear independence, basis, and dimension

1.1 Axioms of a Vector Space

A vector space (or linear space) over a field $\mathbb{F}$ is a set $V$ equipped with two operations—vector addition and scalar multiplication—satisfying ten fundamental axioms. In linear algebra, we typically work over $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$.

Definition: Vector Space

A vector space over a field $\mathbb{F}$ is a set $V$ together with operations $+: V \times V \to V$ and $\cdot: \mathbb{F} \times V \to V$ such that for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\alpha, \beta \in \mathbb{F}$:

A1 (Closure under +): $\mathbf{u} + \mathbf{v} \in V$
A2 (Commutativity): $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
A3 (Associativity): $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
A4 (Zero vector): $\exists\, \mathbf{0} \in V$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$
A5 (Additive inverse): $\forall\, \mathbf{u} \in V,\, \exists\, (-\mathbf{u})$ with $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$
S1 (Closure under scalar mult): $\alpha \mathbf{u} \in V$
S2 (Distributivity over vectors): $\alpha(\mathbf{u} + \mathbf{v}) = \alpha\mathbf{u} + \alpha\mathbf{v}$
S3 (Distributivity over scalars): $(\alpha + \beta)\mathbf{u} = \alpha\mathbf{u} + \beta\mathbf{u}$
S4 (Associativity of scalars): $(\alpha\beta)\mathbf{u} = \alpha(\beta\mathbf{u})$
S5 (Multiplicative identity): $1 \cdot \mathbf{u} = \mathbf{u}$

The axioms are not independent—some follow from others in certain formulations—but this standard list provides a complete characterization. The key insight is that vector spaces abstract the essential algebraic properties of arrows in Euclidean space to a vastly more general setting.

Examples of Vector Spaces

$\mathbb{R}^n$: The space of n-tuples of real numbers with componentwise operations.
$\mathcal{P}_n(\mathbb{R})$: Polynomials of degree at most $n$ with real coefficients.
$C([a,b])$: Continuous real-valued functions on $[a,b]$.
$\mathbb{R}^{m \times n}$: The space of $m \times n$ real matrices.
$\ell^2$: Square-summable sequences $(a_n)$ with $\sum |a_n|^2 < \infty$.

Figure. A 2D vector space with basis vectors e₁ and e₂. Any vector v can be written as a linear combination v = ae₁ + be₂, with dashed projections showing the components.

Derivation 1: Uniqueness of the Zero Vector

We prove that the zero vector is unique. Suppose $\mathbf{0}$ and $\mathbf{0}'$ are both zero vectors. Then:

$$\mathbf{0}' = \mathbf{0}' + \mathbf{0} = \mathbf{0} + \mathbf{0}' = \mathbf{0}$$

The first equality uses $\mathbf{0}$ as a zero vector (A4), and the second uses commutativity (A2), and the third uses $\mathbf{0}'$ as a zero vector (A4). Thus $\mathbf{0} = \mathbf{0}'$.

Derivation 2: $0 \cdot \mathbf{v} = \mathbf{0}$

We derive that multiplying any vector by the scalar zero yields the zero vector:

$$0 \cdot \mathbf{v} = (0 + 0)\mathbf{v} = 0\mathbf{v} + 0\mathbf{v}$$

Adding $-(0\mathbf{v})$ to both sides gives $\mathbf{0} = 0\mathbf{v}$. This result, while seemingly trivial, is not an axiom but a theorem derived from the axioms.

1.2 Subspaces

A subspace of a vector space $V$ is a subset $W \subseteq V$ that is itself a vector space under the same operations. The subspace test provides an efficient way to verify this property.

Subspace Test

A nonempty subset $W \subseteq V$ is a subspace if and only if for all$\mathbf{u}, \mathbf{v} \in W$ and $\alpha \in \mathbb{F}$:

$$\alpha \mathbf{u} + \mathbf{v} \in W$$

This single condition simultaneously guarantees closure under addition (set $\alpha = 1$), closure under scalar multiplication (set $\mathbf{v} = \mathbf{0}$), and the existence of the zero vector (set $\alpha = -1, \mathbf{v} = \mathbf{u}$).

Derivation 3: Intersection of Subspaces

We prove that the intersection of any collection of subspaces is itself a subspace. Let $\{W_i\}_{i \in I}$ be a family of subspaces of $V$, and let$W = \bigcap_{i \in I} W_i$.

First, $\mathbf{0} \in W_i$ for all $i$, so $\mathbf{0} \in W$ and$W \neq \emptyset$. Now let $\mathbf{u}, \mathbf{v} \in W$ and $\alpha \in \mathbb{F}$. For each $i$, since $\mathbf{u}, \mathbf{v} \in W_i$ and $W_i$ is a subspace, we have $\alpha\mathbf{u} + \mathbf{v} \in W_i$. Since this holds for all $i$,$\alpha\mathbf{u} + \mathbf{v} \in W$. By the subspace test, $W$ is a subspace.

Important Subspace Examples

Null space: $\ker(A) = \{\mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = \mathbf{0}\}$ is a subspace of $\mathbb{R}^n$.
Column space: $\text{Col}(A) = \{A\mathbf{x} : \mathbf{x} \in \mathbb{R}^n\}$ is a subspace of $\mathbb{R}^m$.
Solution space: Solutions to a homogeneous system form a subspace.
Not a subspace: The unit circle $\{(x,y) : x^2 + y^2 = 1\}$ is NOT a subspace (doesn't contain $\mathbf{0}$).

1.3 Span and Linear Combinations

A linear combination of vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k \in V$ is any vector of the form:

$$\mathbf{w} = \alpha_1 \mathbf{v}_1 + \alpha_2 \mathbf{v}_2 + \cdots + \alpha_k \mathbf{v}_k = \sum_{i=1}^{k} \alpha_i \mathbf{v}_i$$

where $\alpha_1, \ldots, \alpha_k \in \mathbb{F}$ are scalars.

Definition: Span

The span of a set $S = \{\mathbf{v}_1, \ldots, \mathbf{v}_k\}$ is the set of all linear combinations of vectors in $S$:

$$\text{Span}(S) = \left\{\sum_{i=1}^{k} \alpha_i \mathbf{v}_i : \alpha_i \in \mathbb{F}\right\}$$

The span is always a subspace—in fact, it is the smallest subspace containing $S$.

We say that $S$ spans (or generates) $V$ if$\text{Span}(S) = V$. This means every vector in $V$ can be written as a linear combination of vectors in $S$.

Example: Spanning $\mathbb{R}^3$

The standard basis vectors $\mathbf{e}_1 = (1,0,0)$, $\mathbf{e}_2 = (0,1,0)$,$\mathbf{e}_3 = (0,0,1)$ span $\mathbb{R}^3$ because any vector$(a,b,c)$ can be written as $a\mathbf{e}_1 + b\mathbf{e}_2 + c\mathbf{e}_3$. However, the set $\{(1,0,0), (0,1,0)\}$ does not span $\mathbb{R}^3$since vectors with nonzero third component cannot be represented.

1.4 Linear Independence

Definition: Linear Independence

Vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are linearly independent if the only solution to:

$$\alpha_1 \mathbf{v}_1 + \alpha_2 \mathbf{v}_2 + \cdots + \alpha_k \mathbf{v}_k = \mathbf{0}$$

is the trivial solution $\alpha_1 = \alpha_2 = \cdots = \alpha_k = 0$. Otherwise, the vectors are linearly dependent.

Linear independence captures the idea that no vector in the set is "redundant"—none can be expressed as a linear combination of the others. This is fundamental: if $\mathbf{v}_j = \sum_{i \neq j} \beta_i \mathbf{v}_i$, then $\sum_{i \neq j} \beta_i \mathbf{v}_i - \mathbf{v}_j = \mathbf{0}$ is a nontrivial relation.

Derivation 4: Steinitz Exchange Lemma

The Steinitz exchange lemma states: if $\{\mathbf{v}_1, \ldots, \mathbf{v}_m\}$ is linearly independent and each $\mathbf{v}_i$ is in the span of $\{\mathbf{w}_1, \ldots, \mathbf{w}_n\}$, then $m \leq n$.

The proof proceeds by induction. Since $\mathbf{v}_1 \in \text{Span}(\mathbf{w}_1, \ldots, \mathbf{w}_n)$and $\mathbf{v}_1 \neq \mathbf{0}$ (by linear independence), at least one $\mathbf{w}_j$ has a nonzero coefficient. We can exchange $\mathbf{w}_j$ with $\mathbf{v}_1$ while preserving the span. Repeating this process $m$ times, we replace $m$ of the $\mathbf{w}_j$'s with $\mathbf{v}_i$'s, which requires $m \leq n$.

Key Consequence

The Steinitz exchange lemma implies that any two bases of a finite-dimensional vector space have the same number of elements. This number is the dimension of the space.

1.5 Basis and Dimension

Definition: Basis

A basis for a vector space $V$ is a set $\mathcal{B} = \{\mathbf{e}_1, \ldots, \mathbf{e}_n\}$ that is:

1. Linearly independent: No vector is a linear combination of the others.
2. Spanning: $\text{Span}(\mathcal{B}) = V$.

Equivalently, $\mathcal{B}$ is a basis if every $\mathbf{v} \in V$ can be writtenuniquely as a linear combination of the basis vectors. This unique representation is what makes bases so useful—they provide a coordinate system for the vector space.

Derivation 5: Dimension Theorem for Subspaces

Let $W_1, W_2$ be finite-dimensional subspaces of $V$. We prove:

$$\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)$$

Let $\{\mathbf{u}_1, \ldots, \mathbf{u}_r\}$ be a basis for $W_1 \cap W_2$. Extend to a basis $\{\mathbf{u}_1, \ldots, \mathbf{u}_r, \mathbf{v}_1, \ldots, \mathbf{v}_s\}$of $W_1$ and a basis $\{\mathbf{u}_1, \ldots, \mathbf{u}_r, \mathbf{w}_1, \ldots, \mathbf{w}_t\}$of $W_2$. Then $\{\mathbf{u}_1, \ldots, \mathbf{u}_r, \mathbf{v}_1, \ldots, \mathbf{v}_s, \mathbf{w}_1, \ldots, \mathbf{w}_t\}$is a basis for $W_1 + W_2$, giving:

$$\dim(W_1 + W_2) = r + s + t = (r + s) + (r + t) - r = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)$$

Dimension of Common Spaces

$\dim(\mathbb{R}^n) = n$
$\dim(\mathcal{P}_n) = n + 1$ (polynomials of degree $\leq n$)
$\dim(\mathbb{R}^{m \times n}) = mn$ (matrices)
$\dim(\text{Sym}_n(\mathbb{R})) = \frac{n(n+1)}{2}$ (symmetric matrices)
$\dim(\text{Skew}_n(\mathbb{R})) = \frac{n(n-1)}{2}$ (skew-symmetric matrices)

1.6 Change of Basis

Given two bases $\mathcal{B} = \{\mathbf{e}_1, \ldots, \mathbf{e}_n\}$ and$\mathcal{B}' = \{\mathbf{e}'_1, \ldots, \mathbf{e}'_n\}$ of $V$, the change-of-basis matrix $P$ relates the coordinates:

$$[\mathbf{v}]_{\mathcal{B}'} = P^{-1} [\mathbf{v}]_{\mathcal{B}}$$

where $P$ is the matrix whose columns are the coordinates of the new basis vectors expressed in the old basis: $P = \begin{pmatrix} [\mathbf{e}'_1]_{\mathcal{B}} & \cdots & [\mathbf{e}'_n]_{\mathcal{B}} \end{pmatrix}$.

If a linear map $T: V \to V$ has matrix representation $A$ with respect to$\mathcal{B}$, then its representation with respect to $\mathcal{B}'$ is:

$$A' = P^{-1} A P$$

This is the similarity transformation, and matrices related this way are called similar. Similar matrices represent the same linear map in different coordinate systems.

1.7 Direct Sums

A vector space $V$ is the direct sum of subspaces $W_1$ and$W_2$, written $V = W_1 \oplus W_2$, if:

1. $V = W_1 + W_2$ (every vector is the sum of vectors from each subspace)
2. $W_1 \cap W_2 = \{\mathbf{0}\}$ (the intersection is trivial)

Equivalently, every $\mathbf{v} \in V$ can be written uniquely as$\mathbf{v} = \mathbf{w}_1 + \mathbf{w}_2$ with $\mathbf{w}_i \in W_i$.

Example: $\mathbb{R}^3$ as a Direct Sum

Let $W_1$ be the $xy$-plane and $W_2$ be the $z$-axis. Then $\mathbb{R}^3 = W_1 \oplus W_2$ because every vector $(a,b,c)$ can be uniquely decomposed as $(a,b,0) + (0,0,c)$. Note that $\dim(\mathbb{R}^3) = \dim(W_1) + \dim(W_2) = 2 + 1 = 3$.

The direct sum generalizes to any finite collection of subspaces: $V = W_1 \oplus W_2 \oplus \cdots \oplus W_k$means every vector has a unique decomposition, and the dimension formula becomes:

$$\dim(V) = \sum_{i=1}^{k} \dim(W_i)$$

1.8 Historical Development

The concept of a vector space evolved over more than a century, emerging from disparate mathematical traditions before being unified into the abstract framework we use today.

Hermann Grassmann (1844)

In his revolutionary Ausdehnungslehre (Theory of Extension), Grassmann developed a calculus of "extensive magnitudes" that anticipated modern vector space theory. He defined operations on abstract quantities that obeyed the laws we now recognize as vector space axioms, and even developed the exterior algebra (wedge product). His work was far ahead of its time and largely unappreciated during his lifetime.

Giuseppe Peano (1888)

Peano gave the first axiomatic definition of a vector space in his Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann. His axiom system was essentially the one we use today, making him a pioneer of the abstract algebraic approach to linear algebra.

Stefan Banach and the Polish School (1920s–1930s)

The Lviv school of mathematics, led by Banach, extended vector space theory to infinite-dimensional settings, creating functional analysis. Banach spaces (complete normed vector spaces) became the foundation of modern analysis and quantum mechanics.

Modern Formalization (1940s–1960s)

The Bourbaki group systematized linear algebra in its modern abstract form. Paul Halmos's Finite-Dimensional Vector Spaces (1942) became the standard reference, presenting the subject in the coordinate-free style that emphasizes linear maps over matrices.

1.9 Applications

Computer Graphics and 3D Rendering

Every 3D scene in computer graphics is fundamentally a collection of vectors in $\mathbb{R}^3$(or $\mathbb{R}^4$ in homogeneous coordinates). Rotations, translations, scaling, and perspective projections are all linear (or affine) maps. The GPU performs billions of vector space operations per second to render real-time graphics.

The concept of basis is directly applied when choosing coordinate systems for models, cameras, and lights. Change-of-basis transformations convert between local model coordinates, world coordinates, camera coordinates, and screen coordinates.

Quantum Mechanics

Quantum states live in complex vector spaces (specifically, Hilbert spaces). The superposition principle—that a quantum system can be in a combination of states—is precisely the statement that quantum states form a vector space. Observables are linear operators, and measurements correspond to projections onto subspaces.

A qubit, the basic unit of quantum computing, is an element of $\mathbb{C}^2$, and an n-qubit system lives in the tensor product space $(\mathbb{C}^2)^{\otimes n}$, which has dimension $2^n$—the exponential growth of this dimension is the source of quantum computing's power.

Signal Processing

Signals (audio, images, sensor data) are vectors in high-dimensional spaces. The Fourier transform is a change of basis from the time domain to the frequency domain. Compression algorithms like MP3 and JPEG exploit the fact that most signals have sparse representations in appropriate bases (Fourier, wavelet, DCT), allowing us to discard components with small coefficients.

Error-Correcting Codes

In coding theory, codewords are vectors in $\mathbb{F}_2^n$ (the vector space over the field with two elements). A linear code is a subspace of $\mathbb{F}_2^n$, and error detection and correction exploit the structure of these subspaces. The Hamming code, Reed-Solomon codes, and LDPC codes all rely on linear algebra over finite fields.

Machine Learning

Feature spaces in machine learning are vector spaces where each data point is represented as a vector of features. Dimensionality reduction techniques (PCA, t-SNE) find lower-dimensional subspaces that capture most of the variance. Neural networks compute compositions of affine maps (matrix multiplications plus bias vectors) with nonlinear activations, and the weight space of a neural network is itself a high-dimensional vector space over which we optimize.

1.10 Computational Exploration

The following simulation verifies vector space axioms numerically, tests linear independence using matrix rank, demonstrates basis representation and change of basis, and visualizes span and subspace structure. We also verify the dimension theorem for sums of subspaces.

Vector Spaces: Axioms, Independence, Basis, and Dimension

Python

vector_spaces.py202 lines

import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

# ============================================================
# Vector Spaces: Basis, Dimension, and Linear Independence
# ============================================================

# --- 1. Verify vector space axioms for R^3 ---
print("=" * 60)
print("VECTOR SPACE AXIOMS VERIFICATION IN R^3")
print("=" * 60)

u = np.array([1.0, 2.0, 3.0])
v = np.array([4.0, 5.0, 6.0])
w = np.array([7.0, 8.0, 9.0])
alpha = 2.5
beta = -1.3

print(f"u = {u}")
print(f"v = {v}")
print(f"w = {w}")
print(f"alpha = {alpha}, beta = {beta}")
print()

# Axiom 1: Closure under addition
print("Axiom 1 (Closure +):", u + v)
# Axiom 2: Commutativity
print("Axiom 2 (Commutativity):", np.allclose(u + v, v + u))
# Axiom 3: Associativity of addition
print("Axiom 3 (Associativity +):", np.allclose((u + v) + w, u + (v + w)))
# Axiom 4: Zero vector
zero = np.zeros(3)
print("Axiom 4 (Zero vector):", np.allclose(u + zero, u))
# Axiom 5: Additive inverse
print("Axiom 5 (Additive inverse):", np.allclose(u + (-u), zero))
# Axiom 6: Closure under scalar multiplication
print("Axiom 6 (Closure *):", alpha * u)
# Axiom 7: Distributivity over vectors
print("Axiom 7 (Distrib vectors):", np.allclose(alpha * (u + v), alpha * u + alpha * v))
# Axiom 8: Distributivity over scalars
print("Axiom 8 (Distrib scalars):", np.allclose((alpha + beta) * u, alpha * u + beta * u))
# Axiom 9: Associativity of scalar multiplication
print("Axiom 9 (Assoc *):", np.allclose((alpha * beta) * u, alpha * (beta * u)))
# Axiom 10: Multiplicative identity
print("Axiom 10 (Identity):", np.allclose(1.0 * u, u))

# --- 2. Linear independence check ---
print()
print("=" * 60)
print("LINEAR INDEPENDENCE")
print("=" * 60)

# Set of vectors
v1 = np.array([1, 0, 0])
v2 = np.array([0, 1, 0])
v3 = np.array([0, 0, 1])
v4 = np.array([1, 1, 0])

A_indep = np.column_stack([v1, v2, v3])
A_dep = np.column_stack([v1, v2, v4])

rank_indep = np.linalg.matrix_rank(A_indep)
rank_dep = np.linalg.matrix_rank(A_dep)

print(f"Vectors v1, v2, v3: rank = {rank_indep} (Independent: {rank_indep == 3})")
print(f"Vectors v1, v2, v4: rank = {rank_dep} (Independent: {rank_dep == 3})")
print(f"  v4 = v1 + v2 = {v1 + v2} (linearly dependent!)")

# --- 3. Basis and dimension ---
print()
print("=" * 60)
print("BASIS AND DIMENSION")
print("=" * 60)

# Standard basis for R^3
e1 = np.array([1, 0, 0])
e2 = np.array([0, 1, 0])
e3 = np.array([0, 0, 1])

# Express a vector in this basis
x = np.array([3, -2, 7])
coeffs = np.linalg.solve(np.column_stack([e1, e2, e3]).astype(float), x.astype(float))
print(f"Vector x = {x}")
print(f"In standard basis: {coeffs[0]}*e1 + {coeffs[1]}*e2 + {coeffs[2]}*e3")

# Non-standard basis
b1 = np.array([1, 1, 0])
b2 = np.array([0, 1, 1])
b3 = np.array([1, 0, 1])
B = np.column_stack([b1, b2, b3]).astype(float)
coords = np.linalg.solve(B, x.astype(float))
print(f"In basis (b1,b2,b3): {coords[0]:.4f}*b1 + {coords[1]:.4f}*b2 + {coords[2]:.4f}*b3")
print(f"Verification: {coords[0]*b1 + coords[1]*b2 + coords[2]*b3}")

# --- 4. Subspace verification and span ---
print()
print("=" * 60)
print("SUBSPACES AND SPAN")
print("=" * 60)

# Subspace spanned by v1, v2 in R^3
print("Span(v1, v2) where v1 = [1,0,0], v2 = [0,1,0]:")
print("  This is the xy-plane in R^3")
print("  Dimension = 2")

# Check if a vector lies in the subspace
test_in = np.array([3, 5, 0])
test_out = np.array([3, 5, 1])
A_sub = np.column_stack([v1, v2]).astype(float)

# Use least squares to check membership
_, res_in, _, _ = np.linalg.lstsq(A_sub, test_in.astype(float), rcond=None)
_, res_out, _, _ = np.linalg.lstsq(A_sub, test_out.astype(float), rcond=None)

print(f"  [3,5,0] in Span? Residual norm: {np.linalg.norm(A_sub @ np.linalg.lstsq(A_sub, test_in.astype(float), rcond=None)[0] - test_in):.6f}")
print(f"  [3,5,1] in Span? Residual norm: {np.linalg.norm(A_sub @ np.linalg.lstsq(A_sub, test_out.astype(float), rcond=None)[0] - test_out):.6f}")

# --- 5. Dimension theorem ---
print()
print("=" * 60)
print("DIMENSION THEOREM")
print("=" * 60)

# For V = R^4, with subspaces W1 and W2
# dim(W1 + W2) = dim(W1) + dim(W2) - dim(W1 intersect W2)
W1_basis = np.array([[1,0,0,0], [0,1,0,0], [0,0,1,0]]).T  # 3-dim
W2_basis = np.array([[0,0,1,0], [0,0,0,1], [0,1,0,0]]).T  # 3-dim
combined = np.hstack([W1_basis, W2_basis])
dim_sum = np.linalg.matrix_rank(combined)
dim_W1 = np.linalg.matrix_rank(W1_basis)
dim_W2 = np.linalg.matrix_rank(W2_basis)
dim_intersect = dim_W1 + dim_W2 - dim_sum

print(f"dim(W1) = {dim_W1}")
print(f"dim(W2) = {dim_W2}")
print(f"dim(W1 + W2) = {dim_sum}")
print(f"dim(W1 ∩ W2) = {dim_intersect}")
print(f"Check: {dim_W1} + {dim_W2} - {dim_intersect} = {dim_sum} ✓")

# --- 6. Visualization ---
fig, axes = plt.subplots(1, 3, figsize=(15, 5))

# Plot 1: Vectors in R^2
ax = axes[0]
vectors = np.array([[1, 0], [0, 1], [1, 1], [-1, 2]])
colors = ["#3b82f6", "#0ea5e9", "#06b6d4", "#38bdf8"]
labels = ["e1", "e2", "e1+e2", "-e1+2e2"]
for i, (vec, color, label) in enumerate(zip(vectors, colors, labels)):
    ax.arrow(0, 0, vec[0], vec[1], head_width=0.08, head_length=0.05,
             fc=color, ec=color, linewidth=2)
    ax.annotate(label, xy=(vec[0]*1.15, vec[1]*1.15), fontsize=10,
                color=color, fontweight="bold")
ax.set_xlim(-1.5, 2)
ax.set_ylim(-0.5, 2.5)
ax.set_aspect("equal")
ax.grid(True, alpha=0.3)
ax.set_title("Vectors in R^2", fontsize=12, fontweight="bold")
ax.axhline(y=0, color="white", linewidth=0.5, alpha=0.5)
ax.axvline(x=0, color="white", linewidth=0.5, alpha=0.5)
ax.set_facecolor("#0f172a")

# Plot 2: Span of two vectors
ax = axes[1]
t = np.linspace(-2, 2, 20)
s = np.linspace(-2, 2, 20)
T, S = np.meshgrid(t, s)
span_v1 = np.array([1, 2])
span_v2 = np.array([2, 1])
X = T * span_v1[0] + S * span_v2[0]
Y = T * span_v1[1] + S * span_v2[1]
ax.scatter(X.flatten(), Y.flatten(), c="#3b82f6", alpha=0.3, s=5)
ax.arrow(0, 0, span_v1[0], span_v1[1], head_width=0.15, head_length=0.1,
         fc="#0ea5e9", ec="#0ea5e9", linewidth=2)
ax.arrow(0, 0, span_v2[0], span_v2[1], head_width=0.15, head_length=0.1,
         fc="#38bdf8", ec="#38bdf8", linewidth=2)
ax.set_xlim(-4, 4)
ax.set_ylim(-4, 4)
ax.set_aspect("equal")
ax.grid(True, alpha=0.3)
ax.set_title("Span of Two Vectors", fontsize=12, fontweight="bold")
ax.set_facecolor("#0f172a")

# Plot 3: Dimension comparison
ax = axes[2]
dims = [1, 2, 3, 4, 5]
num_basis_vectors = dims
num_subspaces = [2**d - 1 for d in dims]
ax.bar(dims, num_subspaces, color="#3b82f6", alpha=0.7, edgecolor="#0ea5e9")
ax.set_xlabel("Dimension n", fontsize=11)
ax.set_ylabel("Number of Proper Subspaces", fontsize=11)
ax.set_title("Growth of Subspace Count", fontsize=12, fontweight="bold")
ax.set_facecolor("#0f172a")
ax.tick_params(colors="white")

plt.tight_layout()
plt.savefig("output.png", dpi=150, bbox_inches="tight",
            facecolor="#0f172a", edgecolor="none")
plt.show()
print("\nPlot saved: Vector space visualizations")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Course Overview Linear Maps →