Intro

Linear algebra is the study of linear maps on finite-dimensional vector spaces.

Within this study, it is beneficial not to study just real numbers, but complex numbers as well. We will generalize the concepts of a plane, and ordinary space to $\mathbb R^{n} and\; \mathbb{C}^{n}$, which will then be generalized to a vector space.

Following this, we will consider subspaces, which are to vector spaces, what subsets are to sets, sums, which are to vector spaces what unions are to subsets, and finally direct sums of subspaces, which are analogous to the union of disjoint sets.

Complex numbers were created so that negative numbers could be given square roots, with the chief assumption being that: $i^2 = -1$.

Complex numbers are numbers that can be expressed in the form $a + bi$, or $x + iy,$ where $a,b,x,y$ are real numbers and $i$ is the imaginary unit, where $i^{2} = 1$.

• While $b$ is a real number, $bi$ constitutes an imaginary value.

Complex Addition is defined: $(a + bi) + (c + di) = (a + c) + (b + d)i$ Complex Multiplication is defined: $(a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i$

• In the event that $a \in \mathbb R$, we associate $a + 0i$, with the real number $a$, which indicates that $\mathbb R$ is a subset of $\mathbb{C}$.

• We treat the number $0 + bi$ as $bi$, and also $0 + 1i$, as $i$.

  • The symbol $i$ was first used to represent $\sqrt{-1}$ by Euler in 1777.

Properties of Complex Arithmetic:

Commutativity:

Addition: $\alpha + \beta = \beta + \alpha, \forall \alpha, \beta \in \mathbb{C}$.

Multiplication: $\alpha (\beta) = \beta (\alpha), \forall \alpha, \beta \in \mathbb{C}$.

Assume that we have $\alpha = a + bi$, $\beta = c + di$, where $a, b, c, d \in \mathbb{R}.$

• $\alpha(\beta) = (a + bi)(c + di) =$

  • $(ac - bd) + (ad + bc)i.$

• $\beta(\alpha) = (c + di)(a + bi) =$

  • $(ca - db) + (cb + da)$

We know that from the commutativity of addition that operations such as $(ac - bd) + (ad + bc), (ca - db) + (cb + da)$ are equivalent . The same follows for $ac, ca, bd, db$, from the commutativity of multiplication.

Associativity:

• $(\alpha + \beta) + \lambda = \alpha + (\beta + \lambda), \forall \alpha, \beta, \lambda \in \mathbb{C}$.

• $\alpha(\beta \lambda) = (\alpha \beta)\lambda, \forall \alpha, \beta, \lambda \in \mathbb{C}$.

Identity:

• $1\lambda = \lambda, \forall \lambda \in \mathbb{C}$

Additive Inverse:

• $\forall \alpha \in \mathbb{C}, \exists ! \beta \in \mathbb{C} : \alpha + \beta = 0$.

Multiplicative Inverse:

• $\forall \alpha \in \mathbb{C}, \exists! \beta: \alpha \beta = 1,\; \text{where}\; a \neq 0\;$

Distributive Property:

• $\lambda (\alpha + \beta) = \lambda ( \alpha) + \lambda (\beta), \forall \alpha, \beta, \lambda \in \mathbb{C}$.

Additive Inverse:

Let $\alpha, \beta \in \mathbb{C}$:

• Let $-\alpha$ denote the additive inverse of $\alpha$, thus $\exists! (-\alpha)$, such that $\alpha + (- \alpha) = 0$.

Subtraction of complex numbers is defined like so:

• $\beta - \alpha = \beta + (- \alpha)$.

Division is defined as: $\beta / \alpha = \beta(1 / \alpha)$.

Also, if we let $\alpha \gt 0$, then $1/\alpha$ is the multiplicative inverse of a, such that $\alpha(1 / \alpha) = 1$

Notation: Throughout this book, we will use $\mathbb{F}$, as a stand in for $\mathbb{R}, \text{and}\; \mathbb{C}$, as both of these are examples of fields. What this means, is that if we can prove a theorem holds for $\mathbb{F}$, then we can prove that it holds for $\mathbb{R}, \text{and}\; \mathbb{C}$. The elements of some fields $\mathbb{F}$, are referred to as scalars, which is a fancy term for number. For $\alpha \in \mathbb{F}$, we consider $\alpha^{m}$, to be the product of $\alpha$ with itself $m$ times: $a^{m} = \underbrace{\alpha \cdot \cdot \cdot \alpha}_{\text{m times}}$. From this we can comfortably assume that:

• $(a^{m})^{n} = a^{mn},\forall m,n \in \mathbb{N}\;\text{where}\; m,n \gt 0.$

• $(\alpha \beta)^{n} = \alpha^{n}\beta^{n}, \forall \alpha, \beta \in \mathbb{F}.$

Lists

Before we go into the definition of $\mathbb{R}^{n}$ and $\mathbb{C}^{n}$, we should consider two important examples: The set $\mathbb{R}^{2}$, which can be thought of as a plane, and is the set of all ordered pairs of real numbers, and $\mathbb{R}^{3}$, the set of all points in 3-space:

• $\mathbb{R}^{2} = \{(x,y): x,y \in \mathbb{R} \}$

• $\mathbb{R}^{3} = \{(x,y,z): x,y,z \in \mathbb{R} \}$

In order to generalize these concepts to higher dimensions we need to use the concept of lists:

A list of length $n$, is an ordered collection of $n$ elements: $(x_1 \ldots x_n)$.

• Two lists are equal iff they have the same number of elements in the same order.

Also, in order to define these higher dimensional analogues, we will use $\mathbb{F}$ which is equivalent to $\mathbb{C}\; \text{or}\; \mathbb{R}.$

$\mathbb{F}^{n}$ is the set of all lists of elements of $\mathbb{F}$, where the lists are of length $n$.

We can't visualize $\mathbb R^{n},\; \text{where}\; n \geq 4$, and the same also holds for $\mathbb{C}^{n},\; \text{where}\; n > 1$. However, while these lack physical geometric representations, we can still manipulate elements of $\mathbb{F}^n$ algebraically.

Addition in $\mathbb{F}^{n}$ is defined by adding the corresponding coordinates:

• $(x_{1} \ldots x_n) + (y_1 \ldots y_n) = (x_1 + y_1 \dots x_n + y_n)$

However, it is typically easier to replace these lists with the variable that is subscripted to represent an element of the list, and results in cleaner mathematics.

• If $x, y \in \mathbb{F}^n, x + y = y + x$, meaning commutativity extends to $\mathbb{F}^n$.

Proof of Commutativity of Addition

Let $x = (x_1 \ldots x_n)$, and let $y = (y_1 \ldots y_n)$.

• $x + y = (x_1 \ldots x_n) + (y_1 \ldots y_n) =$

• $(x_1 + y_1 \ldots x_n + y_n) =$

• $(y_1 + x_1 \ldots y_n + x_n) =$

This following line holds, due to the commutativity of addition in $\mathbb{F}$.

• $(y_1 \ldots y_n) + (x_1 \ldots x_n) =$

• $y + x\;\;\;\$ $\blacksquare.$

Chapter 1.A Exercises

Find A Four-Tuple That Satisfies The Equation

Chapter 1.C Exercises

Proof the Union of Subspaces of V is a Subspace of V