Linear Algebra ...Done

Chapter 1

...Vector Spaces

(Read Pages 1-24 in the text)

Parts A & B

... Definition of a Vector Space

Sets

The central objects in mathematics are sets. A set is a collection of things. The things in a set are called elements.

Equality of sets

Two sets \(A\) and \(B\) are equal (they are the same set) if they contain exactly the same elements, that is, the following two statements hold:

If \(x\in A\), then \(x\in B\).                       

If \(x\in B\), then \(x\in A\).                       

In this case we write \(A=B\).

Important example:

Important example: Consider the sets \(A=\{1,2,2,3\}\) and \(B=\{1,2,3\}\). We can easily see that every element of \(A\) is an element of \(B\), and every element of \(B\) is an element of \(A\). Thus

\[\{1,2,2,3\}=\{1,2,3\}.\]

An element is either in a set or it is not. Sets cannot contain multiple copies of an element.

(\(\leftarrow\)Also written \(A\subset B\).)

(\(\leftarrow\)Also written \(B\subset A\).)

Definition 1.8.

• Suppose \(n\) is a nonnegative integer. A list of length \(n\) is an ordered collection of \(n\) elements.
• Two lists are equal if and only if they have the same length and the same elements in the same order.

Lists

Example.

• The list \((3,1,-2)\) is a list of length \(3\).
• The lists \((1,2,2,3)\) and \((1,2,3)\) are not equal since they have different lenghts.
• The lists \((1,2,2,3)\) and \((1,2,3,2)\) are not equal. They have the same length and the same elements, but not in the same order.

To contrast with sets, lists are often written as elements separated by commas and surrounded by parentheses.

Notation 1.6

Throughout this course \(\mathbb{F}\) stands for either \(\mathbb{R}\) or \(\mathbb{C}\).

(In the textbook they use bold letters \(\textbf{F,R,C}\) instead of the "blackboard bold" font.)

Definition 1.11

Given a positive integer \(n\), the symbol \(\mathbb{F}^{n}\) denotes the set of all list of length \(n\) in \(\mathbb{F}\). That is,

\[\mathbb{F}^{n} = \{(x_{1},\ldots,x_{n}) : x_{k}\in\mathbb{F}\text{ for } k=1,\ldots,n\}.\]

For \((x_{1},\ldots,x_{n})\in\mathbb{F}^{n}\) and \(k\in\{1,\ldots,n\}\), we say that \(x_{k}\) is the \(k^{\text{th}}\) coordinate of \((x_{1},\ldots,x_{n})\).

The set \(\mathbb{F}^{n}\)

Example. The set \(\mathbb{R}^{2}\) consists of all lists of real numbers of length 2. Thus, \((0,1)\), \((-2,\frac{1}{3})\), and \((\frac{\pi}{3},\frac{\sqrt{3}}{2})\) are elements of \(\mathbb{R}^{2}\).

Algebra in \(\mathbb{F}^{n}\)

Definition 1.13

Addition in \(\mathbb{F}^{n}\) is defined by adding corresponding coordinates, that is,

\[(x_{1},\ldots,x_{n}) + (y_{1},\ldots,y_{n}) = (x_{1}+y_{1},\ldots,x_{n}+y_{n}).\]

Theorem 1.14. (Commutativity of addition in \(\mathbb{F}^{n}\))

If \(x,y\in\mathbb{F}^{n}\), then \(x+y = y+x\).

Proof. (on the board and in the text)

More algebra in \(\mathbb{F}^{n}\)

Definition 1.17

For \(x\in\mathbb{F}^{n}\), the additive inverse of \(x\), denoted by \(-x\), is the list with the property that

\[x + (-x) = 0\]

Hence, if \(x = (x_{1},\ldots,x_{n})\), then \(-x = (-x_{1},\ldots,-x_{n})\).

Definition 1.18

The product of a number \(\lambda\in\mathbb{F}\), and a list \(x = (x_{1},\ldots,x_{n})\in\mathbb{F}^{n}\) is computed by multiplying each coordinate of \(x\) by \(\lambda\), that is

\[\lambda x = \lambda(x_{1},\ldots,x_{n}) = (\lambda x_{1},\ldots,\lambda x_{n}).\]

Let's generalize \(\mathbb{F}^{n}\)

Definition 1.19

• An addition on a set \(V\) is a function that assigns an element \(u+v\in V\) to each pair \(u,v\in V\)
• A scalar multiplication on a set \(V\) is a function that assigns an element \(\lambda v\in V\) to each \(\lambda\in \mathbb{F}\) and each \(v\in V\)

We need a set, call it \(V\), and a way to add and scalar multiply...

We want the things in \(V\) to behave like vectors...

Definition 1.20 A vector space is a set \(V\) along with an addition and scalar multiplication on \(V\) with the following properties:

\(\bullet\) Commutativity \[u+v=v+u\quad\text{for all }u,v\in V.\]

\(\bullet\) Associativity of addition \[(u+v)+w=v+(u+w)\quad\text{for all }u,v,w\in V.\]

\(\bullet\) Associativity of scalar multiplication \[(ab)v=a(bv)\quad\text{for all }v\in V\text{ and }a,b\in\mathbb{F}.\]

\(\bullet\) Additive identity \[\exists\, 0\in V\text{ such that }0+v=v\quad\text{for all }v\in V.\]

\(\bullet\) Additive inverse \[\forall\,v\in V,\ \exists w\in V,\ v+w=0.\]

\(\bullet\) Multiplicative identity\[1v=v\quad\text{for all }v\in V.\]

\(\bullet\) Distributive properties: For all \(u,v\in V\) and \(a,b\in\mathbb{F}\), \[a(u+v) = au+av\quad\text{and}\quad (a+b)v = av+bv.\]

Examples of Vector Spaces

Example 1. For any positive integer \(n\), the set \(\mathbb{F}^{n}\) (with addition and multiplication as defined previously) is a vector space.

Terminology:

A vector space \(V\) comes with a field \(\mathbb{F}\) (either \(\mathbb{R}\) or \(\mathbb{C}\)).

• We say that \(V\) is a vector space over \(\mathbb{F}\).
• The field \(\mathbb{F}\) is called the field of scalars.
• If \(V\) is a vector space over \(\mathbb{R}\), then we call \(V\) a real vector space.
• If \(V\) is a vector space over \(\mathbb{C}\), then we call \(V\) a complex vector space​.

Thus, the vector space \(\mathbb{R}^{n}\) is a real vector space, and \(\mathbb{C}^{n}\) is a complex vector space.

Examples of Vector Spaces

Example 2. The set \(\mathbb{F}^{\infty}\) is the set of all sequences of elements of \(\mathbb{F}\), that is,

\[\mathbb{F}^{\infty} = \{(x_{1},x_{2},\ldots) : x_{k}\in\mathbb{F}\text{ for }k=1,2,\ldots\}.\]

Defind addition and scalar multiplication by

\[(x_{1},x_{2},\ldots) + (y_{1},y_{2},\ldots) = (x_{1}+y_{1},x_{2},y_{2},\ldots)\]

and

\[\lambda(x_{1},x_{2},\ldots) = (\lambda x_{1},\lambda x_{2},\ldots).\]

This is a vector space over \(\mathbb{F}\), and the additive identity (zero vector) is the sequence of all \(0\)'s.

Examples of Vector Spaces

Example 3. Let \(S\) be any nonempty set. The set \(\mathbb{F}^{S}\) denotes the set of all functions from \(S\) to \(\mathbb{F}\).

Define addition and scalar multiplication on \(\mathbb{F}^{S}\) as follows:

For \(f,g\in\mathbb{F}^{S}\), define the function \(f+g: S\to \mathbb{F}\) by

\[(f+g)(x) = f(x)+g(x)\quad\text{for all }x\in S.\]

For \(f\in\mathbb{F}^{S}\) and \(\lambda\in \mathbb{F}\), define the function \(\lambda f: S\to \mathbb{F}\) by

\[(\lambda f)(x) = \lambda f(x)\quad\text{for all }x\in S.\]

We see that \(\mathbb{F}^{S}\) is a vector space over \(\mathbb{F}\).

Moreover, the zero vector is the zero function, that is, the function \(0:S\to\mathbb{F}\) given by

\[0(x) = 0\quad\text{for all }x\in S,\]

and for \(f\in\mathbb{F}^{S}\), the additive inverse is the function \(-f:S\to\mathbb{F}\) given by

\[(-f)(x) = -f(x)\quad\text{for all }x\in S.\]

Notation 1.29: For the rest of this course, the symbol \(V\) denotes a vector space over \(\mathbb{F}\).

We know that \(V\) contains an additive identity, but could it contain two different ones?

More deductions about vector spaces

Suppose \(v\in V\). What is \(1v\)?

Part C

... Subspaces

Subsets

Recall that for two sets \(A\) and \(B\), we write \(A\subset B\) (or \(A\subseteq B\), they both mean the same thing) to mean the following:

"If \(x\) is an element of \(A\), then \(x\) is an element of \(B\)."

Example. Define the set

\[A = \{(x,x^2) \in\mathbb{R}^{2} : x\in\mathbb{R}\}.\]

It is clear that \(A\subset \mathbb{R}^2\). However, \(A\) is not closed under addition or scalar multiplication in \(\R^2\). Indeed, note that \((1,1)\) and \((2,4)\) are both elements of \(A\), but

\[(1,1)+(2,4) = (3,5)\notin A\]

and 

\[(-1)(2,4) = (-2,-4)\notin A.\]

Definition 1.33

A subset \(U\) of \(V\) is called a subspace of \(V\) if \(U\) is also a vector space with the same additive identity, addition, and scalar multiplication as on \(V\).

Example. Define the set

\[U = \{(x,x) \in\mathbb{R}^{2}: x\in\mathbb{R}\}.\]

All of the properties of a vector space follow easily once we notice that given two vectors \((x,x)\) and \((y,y)\) in \(U\), and a scalar \(c\in\mathbb{R}\) we have

\[(x,x)+(y,y) = (x+y,x+y)\in U,\]

and

\[c(x,x) = (cx,cx)\in U.\]

Thus, we see that \(U\) is a subspace of \(\mathbb{R}^{2}\). 

Theorem 1.34

A subset \(U\) of \(V\) is a subspace if and only if the following three conditions hold:

• (Additive identity) \(0\in U\)
• (Closed under addition) \(u,w\in U\) implies \(u+w\in U\)
• (Closed under scalar mult.) \(u\in U\) and \(c\in\mathbb{F}\) implies \(cu\in U\).