Contents

• Vector spaces
  • Fields—what can we multiply a vector with?
  • An example of a vectorspace: $\mathbb F^n$
• Vector Space Axioms
  • Some first proofs
  • More examples
• Linear subspaces
• Linear combinations
• Solving linear systems of equations
• Linear independence, Bases, and Dimension
  • Components of a vector with respect to a basis
  • Examples—bases for $\R^2$
  • Examples—dimension depends on the number field $\mathbb{F}$
  • Examples—bases for spaces of polynomials

Vector spaces

A vector space is a set of mathematical objects that we would like to call "vectors." These "vectors" can be added to each other, and they can be multiplied with "numbers."

To specify a vector space we therefore have to say

• what the set of vectors $V$ is
• how to define the sum $x+y$ of two vectors $x,y\in V$
• what we mean by "numbers" (there is a choice)
• how to define the product $ax$ of a number $a$ and a vector $x$ .

Fields—what can we multiply a vector with?

In any theory of vectors we would define $2v$ for any vector $v$ by saying $2v = v+v$. But would we have to define $\sqrt{2}\, v$, or $\pi v$, or $(3-i\sqrt3)v$?

What kind of numbers do we allow in our theory of vectors?

The full answer to this question is that we get to choose what we call a number. The only restriction is that the set of numbers we use should form a "field" $\mathbb{F}$, a mathematical concept which is itself defined in terms of axioms that you can find in Appendix C of the textbook.

In this course we will ignore the field axioms and avoid this level of generality. Instead we will always assume the number field $\mathbb F$ is one of the following three examples:

• the rational numbers $\mathbb Q$: a rational number is a number of the form $\frac{p}{q}$ where $p,q$ are integers, and where $q\neq 0$. Two rational numbers $p/q$ and $m/n$ are equal if $pn=mq$.
• the real numbers $\R$: it's a long story. Take math 421 and/or 521 to get a fuller description of the real numbers. We'll assume that they are familiar from calculus and not worry about questions like “do infinitely small number exist?”.
• or the complex numbers $\mathbb C$: numbers of the form $a+bi$ where $a,b\in\R$ and where $i^2=-1$.

Note: the notation $\R$, $\mathbb{Q}$, $\mathbb{C}$ has been completely standard since the 1980ies, but our textbook was written in 1979 and writes $R$, ${Q}$, and $C$ instead.

Another note: number and scalar mean the same thing for Friedberg-Insel-Spence.

An example of a vectorspace: $\mathbb F^n$

$\mathbb{F}^n$ consists of all $n$-tuples $(x_1, \dots, x_n)$ where $x_1, \dots, x_n\in\mathbb F$. Once we start using matrices it turns out be more convenient to change notation, and write the components of $(x_1, \dots, x_n)$ in a column:

$(x_1, \dots, x_n) = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}.$

Addition and scalar multiplication are defined by

$\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} +\begin{pmatrix} y_1 \\ \vdots \\ y_n \end{pmatrix} =\begin{pmatrix} x_1+y_1 \\ \vdots \\ x_n+y_n \end{pmatrix},\qquad a\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} =\begin{pmatrix} a\,x_1 \\ \vdots \\ a\,x_n \end{pmatrix}.$

The standard basis vectors are

$\mathbf{e}_1=\begin{pmatrix} 1 \\0 \\ \vdots\\0 \end{pmatrix},\quad \mathbf{e}_2=\begin{pmatrix} 0 \\1 \\ \vdots\\0 \end{pmatrix},\quad\dots\quad \mathbf{e}_n=\begin{pmatrix} 0\\0 \\ \vdots\\1 \end{pmatrix}.$

Every vector $x=(x_1, \dots, x_n)$ can be written as a linear combination of the standard basis vectors:

$(x_1, \dots, x_n) = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} =x_1\mathbf{e}_1 + x_2\mathbf{e}_2 + \cdots + x_n\mathbf{e}_n$

For example, if

$x=\begin{pmatrix} \frac32 \\[6pt] \frac{22}{7} \end{pmatrix},\quad y=\begin{pmatrix} \frac32 \\[6pt] \pi \end{pmatrix},\quad z=\begin{pmatrix} 1 \\ i \end{pmatrix},\quad$

then

$x\in \mathbb{Q}^2, x\in\R^2, x\in\mathbb{C}^2,\qquad y\in\mathbb{C}^2, y\in\R^2, y\not\in\mathbb{Q}^2,\qquad z\in\mathbb{C}^2, z\not\in\R^2, z\not\in\mathbb{Q}^2.$

Vector Space Axioms

The axioms themselves are fairly standard, but the numbering differs from book to book. In our text book the authors wrote the axioms like this:

• (VS1) For all $x,y\in V$ one has $x+y=y+x$
• (VS2) For all $x,y,z\in V$ one has $(x+y)+z = x+(y+z)$
• (VS3) There is a $0_V\in V$ such that for all $x\in V$ one has $x+0_V=x$
• (VS4) For each $x\in V$ there is a $y\in V$ such that $x+y=0_V$
• (VS5) For each $x\in V$ one has $1x=x$
• (VS6, 7, 8) For all $a,b\in\mathbb F$, and all $x, y\in V$ one has

  $(ab)x = a(bx), \quad a(x+y) = ax+ay, \quad (a+b)x = ax+bx.$

Some first proofs

The following facts are consequences of the axioms (VS1–8):

• $V=\R^n$ and $\mathbb{F}=\R$ satisfy the vector space axioms (VS1–8).
• there is only one zero vector in $V$
• For each $x\in V$ there is only one additive inverse "$-x$"
• The cancellation theorem (Theorem 1.1 in the book)
• $0_{\mathbb F}\, x=0_V$ for all $x\in V$.
• $a\, 0_V=0_V$ for all $a\in\mathbb F$
• $(-1)x = -x$ for all $x\in V$

More examples

• $\mathsf{M}_{m\times n}(\mathbb{F})$: all $m\times n$ matrices with entries from $\mathbb F$
• $\mathsf P(\mathbb{F})$: all polynomials with coefficients in $\mathbb{F}$
• $\mathsf P_n(\mathbb{F})$: all polynomials with coefficients in $\mathbb{F}$ of degree $\leq n$
• $\mathsf{T}(\mathbb{F})$: all trigonometric polynomials with coefficients in $\mathbb{F}$
• $\mathsf{T}_n(\mathbb{F})$: all trigonometric polynomials with coefficients in $\mathbb{F}$ of degree $\leq n$
• $\mathbb C$ as a real vector space
• $\R$ as a vectorspace over $\mathbb Q$
• $\mathcal{F}([a,b])$: all real valued functions defined on an interval $[a,b]\subset\R$
• $\mathcal{C}([a,b])$: all continuous real valued functions defined on an interval $[a,b]\subset\R$
• $\mathcal{C}^\infty(a,b)$: all infinitely often differentiable real valued functions that are defined on the interval $(a,b)\subset\R$.

Linear subspaces

Terminology: a linear subspace is the same thing as a subspace

Instead of using the definition in the book we will take the following equivalent definition of a (linear) subspace $W$ of a vectorspace $V$

Definition. If $V$ is a vector space over some field $\mathbb{F}$ then a subset $W\subset V$ is called a subspace if

• $W$ is not empty, and
• for all $x,y\in W$ one has $x+y\in W$, and
• for all $x\in W$ and $a\in\mathbb{F}$ one has $ax\in W$.

Properties of linear subspaces

If $W$ is a linear subspace of the vector space $V$, and $\mathbb F$ is the number field, then:

1. $0_V\in W$
2. for each $x\in W$ one has $-x\in W$
3. $W$ with the addition and scalar multiplication from $V$ is a vector space over $\mathbb{F}$

Examples of linear subspaces

• a line through the origin—other lines?
• the $xy$ plane in $\R^3$
• symmetric matrices $\subset\mathsf M_{n\times n}(\mathbb{F})$
• $\mathsf{P}_n(\mathbb{F})$ : polynomials of degree at most $n$

Solutions to a linear differential equation

Let $V=\mathcal{F}(\R,\R)$ be the set of all functions $f:\R\to\R$. Then $(V,\R)$ is a vector space.

Let $W\subset V$ be the set of functions $f:\R\to\R$ that satisfy the differential equation

$f''(x)−4f(x)=0 \text{ for all $x\in\R$}$

Problem: Prove that $W$ a linear subspace of $V$.

Let $U\subset V$ be the set of functions $f:\R\to\R$ that satisfy the differential equation

$f''(x)−4f(x)=16 \text{ for all $x\in\R$}$

Question: is $U$ a linear subspace of $V$?

Theorem (1.4). If $W_1, \dots, W_n\subset V$ are linear subspaces, then $W = W_1\cap W_2\cap \cdots \cap W_n$ is also a linear subspace.

Linear combinations

Definition. A linear combination of vectors $v_1, v_2, \dots, v_n\in V$ is any vector of the form $a_1v_1+\cdots+a_nv_n$, for any choice of numbers $a_1, \dots, a_n\in\mathbb F$.

Definition. The span of a set of vectors $S\subset V$ is the set of all linear combinations of vectors in $S$.

$\mathrm{span}(S) = \left\{a_1v_1+\cdots+a_nv_n \mid a_1, \dots, a_n\in\mathbb F, v_1, \dots, v_n\in V, n\geq 0\right\}$

Theorem. For any subset $S\subset V$ of a vector space $V$, $\mathrm{span}(S)$ is a linear subspace of $V$.

Theorem. If $W\subset V$ is a linear subspace of $V$ and if $S\subset W$, then $\mathrm{span}(S)\subset W$.

Definition. If $S\subset V$ and if $\mathrm{span}(S)=V$ then "$S$ spans $V$", or, "$S$ generates $V$."

Solving linear systems of equations

The book describes the standard method of solving a system of $n$ equations with $m$ unknowns.

Linear independence, Bases, and Dimension

Definition of independence. A set of vectors $\{u_1, \dots, u_n\}\subset V$ is linearly independent if for any $a_1,\dots, a_n\in\mathbb{F}$ one has

$a_1u_1+\cdots+a_nu_n = 0 \implies a_1=a_2=\cdots=a_n=0.$

More generally a set of vectors $\beta\subset V$ is linearly independent if every finite subset $\{u_1, \dots, u_n\}\subset \beta$ is linearly independent. This second definition allows for the possibility that the set $\beta$ is infinite.

Definition of basis. A set of vectors $\{u_1, \dots, u_n\}\subset V$ is a basis for $V$ if

• $\{u_1, \dots, u_n\}$ is linearly independent, and
• $\{u_1, \dots, u_n\}$ spans $V$.

Extension Theorem for Independent Sets. If $\{u_1, \dots, u_n\}\subset V$ is linearly independent, and $v\in V$ is not one the vectors $u_1, \dots, u_n$, then $v\in\mathrm{span}(u_1, \dots, u_n)$ if and only if $\{u_1, \dots, u_n, v\}$ is dependent.

Dimension Theorem. If $\{v_1, \dots, v_m\}\subset \mathrm{span}( u_1, \dots, u_n)$ and if $\{v_1, \dots, v_m\}$ is linearly independent then $m\leq n$.

Corollaries to the Dimension Theorem.

1. If $\{u_1, \dots, u_n\}$ and $\{v_1, \dots, v_m\}$ both are bases of a vector space $V$, then $m=n$.

2. If $L\subset V$ is a linear subspace and $V$ is finite dimensional then $\dim L\leq \dim V$. If $\dim L=\dim V$ then $L=V$.

3. If $L\subset V$ is a linear subspace and $V$ is finite dimensional, and if $\{v_1, \dots, v_k\}\subset L$ is a basis for $L$, then there exist vectors $v_{k+1}, \dots, v_n\in V$ such that $\{v_1, \dots, v_k, v_{k+1}, \dots, v_n\}$ is a basis for $V$.

Definition.

• If a vector space $V$ has a basis $\{u_1, \dots, u_n\}$ with $n$ elements, then $n$ is the dimension of $V$.
• If a vector space $V$ has a basis with finitely many vectors then $V$ is called finite dimensional.

The Dimension Theorem and its corollary imply that the dimension as defined above does not depend on which basis of $V$ you consider.

Components of a vector with respect to a basis

If $\{u_1, \dots, u_n\}$ is a basis for a vector space $V$ and $x\in V$ is any other given vector in $V$, then there exist numbers $x_1, \dots, x_n\in \mathbb{F}$ such that

$x = x_1u_1+\cdots+x_nu_n.$

The numbers $x_1, \dots, x_n$ are called the components, or coefficients, of the vector $x$ with respect to the basis $\{u_1, \dots, u_n\}$.

The coefficients $x_1, \dots, x_n$ are completely determined by the vector $x$ and the basis $\{u_1, \dots, u_n\}$. Namely, if

$x_1'u_1+\cdots+x_n'u_n = x_1u_1+\cdots+x_nu_n$

then we could subtract $x_1u_1$, …, $x_nu_n$ from both sides and we would end up with

$(x_1'-x_1)u_1+\cdots+(x_n'-x_n)u_n = 0.$

Since $u_1, \dots, u_n$ are independent, this implies $x_1=x_1'$, … , $x_n=x_n'$.

Examples—bases for $\R^2$

• $\{\mathbf{e}_1,\mathbf{e}_2\}$ is a basis for $\R^2$
• $\begin{pmatrix}1\\2\end{pmatrix}, \begin{pmatrix}2\\1\end{pmatrix}$ is a basis for $\R^2$

Examples—dimension depends on the number field $\mathbb{F}$

Consider $\mathbb{C}$ as vector space over the complex numbers. It has a basis with one “vector,” namely $\{1\}$.

Now consider $\mathbb{C}$ as vector space over the real numbers. Then the set $\{1, i\}$ is linearly independent, and spans $\mathbb{C}$, i.e. $\{1,i\}$ is a basis for $\mathbb{C}$ over the real numbers.

Consider $V=\R$ as vector space over the rational numbers $\mathbb{Q}$. Then $\{1, \sqrt2\}$ is linearly independent.

Examples—bases for spaces of polynomials

$\{1, x, x^2, x^3, \dots\}$ is a basis for $\mathcal{P}(\R)$. $\mathcal{P}(\R)$ is infinite dimensional.

$\{1, x, x^2\}$ is a basis for $\mathcal{P}_2(\R)$

If we define

$p_0(x) = \frac{(x-1)(x-2)}{(-1)(-2)},\qquad p_1(x) =\frac{x(x-2)}{1\cdot(-1)},\qquad p_2(x) =\frac{x(x-1)}{2\cdot1}$

then $\{p_0(x), p_1(x), p_2(x)\}$ is a basis for $\mathcal{P}_2(\R)$

Given a quadratic polynomial $f(x)\in \mathcal{P}_2(\R)$ what are the coefficients in

$f(x) = c_0p_0(x)+c_1p_1(x)+c_2p_2(x) \quad ??$