[Axler] Chapter 1

Summary of Linear Algebra Done Right: Chapter 1 - Vector Spaces

This summary is based on the definitions and main results presented in Sheldon Axler's Linear Algebra Done Right, 3rd Edition, Chapter 1, which introduces the core object of study: the vector space.

Section 1.A: $\mathbf{R}^n$ and $\mathbf{C}^n$

Section 1.A lays the foundational framework by defining the complex number system and generalizing the familiar Euclidean spaces to $\mathbf{F}^n$, where $\mathbf{F}$ represents a field.

Complex Numbers (Definition 1.1)

A complex number is an ordered pair $(a, b)$, where $a, b \in \mathbf{R}$, written as \(a + bi\). The set of all complex numbers is denoted by \(\mathbf{C} = \{a + bi \mid a, b \in \mathbf{R}\}\). Addition and multiplication in \(\mathbf{C}\) are defined as follows [1]:

• Addition: \((a + bi) + (c + di) = (a + c) + (b + d)i\)
• Multiplication: \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\)

The set of real numbers \(\mathbf{R}\) is considered a subset of \(\mathbf{C}\) by identifying \(a \in \mathbf{R}\) with \(a + 0i \in \mathbf{C}\).[1]

The Field $\mathbf{F}$ and $\mathbf{F}^n$

In this text, $\mathbf{F}$ denotes the field of scalars, which is either the set of real numbers, $\mathbf{R}$, or the set of complex numbers, $\mathbf{C}$.[1]

The set \(\mathbf{F}^n\) is defined as the set of all lists (ordered tuples) of length \(n\) with entries in \(\mathbf{F}\). An element \(x \in \mathbf{F}^n\) is written as \(x = (x_1, \ldots, x_n)\), where \(x_j \in \mathbf{F}\).[2]

Vector Operations in $\mathbf{F}^n$

The fundamental operations on vectors in \(\mathbf{F}^n\) are defined as coordinate-wise:

• Vector Addition: If \(x = (x_1, \ldots, x_n)\) and \(y = (y_1, \ldots, y_n)\) are in \(\mathbf{F}^n\), then: $$\mathbf{x} + \mathbf{y} = (x_1 + y_1, \ldots, x_n + y_n)$$
• Scalar Multiplication: If \(\lambda \in \mathbf{F}\) and \(x = (x_1, \ldots, x_n)\) is in \(\mathbf{F}^n\), then: $$\lambda \mathbf{x} = (\lambda x_1, \ldots, \lambda x_n)$$

Section 1.B: Definition of Vector Space

Section 1.B provides the abstract definition of a vector space, generalizing the properties observed in $\mathbf{F}^n$.[3]

Vector Space Axioms

A vector space is a set $V$ (whose elements are called vectors) along with a field $\mathbf{F}$ (whose elements are called scalars).[1] It must be equipped with two operations: vector addition and scalar multiplication, which satisfy eight core axioms.[4]

Note: The specific list of the eight axioms defining a vector space is not detailed in the available summary materials for this report.[1, 2] In general, these axioms ensure properties like associativity and commutativity of vector addition, the existence of an additive identity (zero vector) and additive inverse, and the distributive and associative properties for scalar multiplication.[4]

Section 1.C: Subspaces

Subspaces are subsets of a vector space that are themselves vector spaces under the same operations.[2]

Subspace Conditions

A subset \(U\) of a vector space \(V\) is a subspace of \(V\) if and only if \(U\) satisfies the following three conditions [5]:

1. Additive Identity: The zero vector \(0\) is in \(U\): \(0 \in U\).
2. Closed under Addition: For all vectors \(u, w \in U\), their sum \(u + w\) is also in \(U\).
3. Closed under Scalar Multiplication: For all scalars \(\lambda \in \mathbf{F}\) and all vectors \(u \in U\), the scalar multiple \(\lambda u\) is also in \(U\).

Sums of Subspaces (Definition 1.36 and Theorem 1.39)

Suppose \(U_1, \ldots, U_m\) are subsets of a vector space \(V\). The sum of these subsets, denoted \(U_1 + \cdots + U_m\), is the set of all possible sums of elements from each subset [2]:

$$U_1 + \cdots + U_m = \{u_1 + \cdots + u_m \mid u_j \in U_j \text{ for each } j\}$$

If \(U_1, \ldots, U_m\) are all subspaces of \(V\), then their sum \(U_1 + \cdots + U_m\) is the smallest subspace of \(V\) containing \(U_1, \ldots, U_m\).[2]

Direct Sums (Definition 1.40 and Theorem 1.44)

The sum \(U_1 + \cdots + U_m\) of subspaces is called a direct sum, denoted \(U_1 \oplus \cdots \oplus U_m\), if every element in the sum can be written in only one way as a sum \(u_1 + \cdots + u_m\), where each \(u_j \in U_j\).[2]

A key condition for the direct sum is provided by Theorem 1.44:

The sum \(U_1 + \cdots + U_m\) is a direct sum if and only if the only way to write the zero vector \(0\) as a sum \(u_1 + \cdots + u_m\), with each \(u_j \in U_j\), is by taking each \(u_j\) equal to \(0\).[2]

Direct Sum of Two Subspaces (Theorem 1.45)

For the case of only two subspaces, $U$ and $W$, the condition simplifies: The sum \(U + W\) is a direct sum if and only if their intersection contains only the zero vector [2]:

$$U \cap W = \{0\}$$

Selected Exercises from Chapter 1: Vector Spaces

These problems are structured to ensure efficient learning by immediately testing the abstract definitions presented in the chapter.

Section 1.A: $\mathbf{R}^n$ and $\mathbf{C}^n$

These exercises verify mastery of complex number arithmetic and the properties of vector operations in $\mathbf{F}^n$.

Exercise 1.A.1: Multiplicative Inverse in $\mathbf{C}$

Suppose $a$ and $b$ are real numbers, not both $0$. Find real numbers $c$ and $d$ such that the multiplicative inverse of $a + bi$ is $c + di$. That is, show that: $$\frac{1}{a + bi} = c + di$$ Express $c$ and $d$ explicitly in terms of $a$ and $b$. This confirms that $\mathbf{C}$ is a field where division (by non-zero elements) is always possible.

Exercise 1.A.2: Vector Addition Properties

Let $\mathbf{F}^3$ be the set of all lists of length three with entries in $\mathbf{F}$ (where $\mathbf{F}$ is $\mathbf{R}$ or $\mathbf{C}$). The vector addition in $\mathbf{F}^3$ is defined as: $$(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1+y_1, x_2+y_2, x_3+y_3)$$ Prove the **Associativity** property of vector addition for all vectors $u, v, w \in \mathbf{F}^3$: $$u + (v + w) = (u + v) + w$$ Explain how this proof relies on the corresponding property of scalar addition in the field $\mathbf{F}$.

Section 1.B: Definition of Vector Space

The goal of this section is to understand the eight axioms. The most instructive exercises are those that demonstrate failure, forcing you to check every axiom.

Exercise 1.B.1: Axiom Failure Test (Closure)

Let $V = \mathbf{R}^2$. Define the standard vector addition, but redefine scalar multiplication $(\odot)$ as follows: $$\lambda \odot (x, y) = (\lambda x, 0)$$ Show that the set $V$ with this modified scalar multiplication fails to satisfy at least one of the eight vector space axioms. Specifically, identify the axiom that fails and provide a counterexample using a scalar $\lambda$ and a vector $v \in V$.

Exercise 1.B.2: Axiom Failure Test (Identity)

Consider the set $W = \{(x, y) \in \mathbf{R}^2 \mid x \ge 0 \text{ and } y \ge 0\}$, which is the first quadrant of the Cartesian plane. Assume standard vector addition and scalar multiplication. Does $W$ satisfy the vector space axiom regarding the existence of the additive inverse? If not, provide a vector $u \in W$ that does not have an additive inverse in $W$.

Section 1.C: Subspaces

These problems test the three necessary and sufficient conditions for a subset to be a subspace, and the definitions of sums and direct sums.

Exercise 1.C.1: The Three Subspace Conditions

Let $V$ be the vector space of all real-valued functions defined on the interval $$, denoted $\mathbf{R}^{}$. Consider the subset $U$ defined as the set of continuous real-valued functions on $$. Prove that $U$ is a subspace of $V$ by verifying the following three conditions: [1, 2]

1. The zero vector $0$ is in $U$.
2. $U$ is closed under vector addition.
3. $U$ is closed under scalar multiplication.

Exercise 1.C.2: Subspace Intersection and Union

Let $U$ and $W$ be two subspaces of a vector space $V$.

1. Prove that the intersection $U \cap W$ is always a subspace of $V$.
2. Prove that the union $U \cup W$ is a subspace of $V$ **if and only if** $U \subseteq W$ or $W \subseteq U$. (This is a classic problem demonstrating why the union of subspaces is not generally a subspace.)

Exercise 1.C.3: Direct Sum Condition

Let $V = \mathbf{R}^3$. Define two subspaces: $$U = \{(x, y, 0) \in \mathbf{R}^3 \mid x, y \in \mathbf{R}\} \quad \text{ (the } xy \text{-plane)}$$ $$W = \{(0, z, w) \in \mathbf{R}^3 \mid z, w \in \mathbf{R}\} \quad \text{ (the } zw \text{-plane)}$$

1. Find the sum $U + W$.
2. Determine whether the sum $U + W$ is a **direct sum**, $U \oplus W$. Justify your answer by calculating the intersection $U \cap W$ and applying the necessary condition for a direct sum of two subspaces. [3]