Looking for context for a textbook quote

[mcastillo356]

TL;DR Summary

The motivation of this post is to know a little about the laws of arithmetic and the rules of linearity for finite sums mentioned in the quotation

Hi, PF, there goes the quote from Calculus 7th ed. by Robert A. Adams and Christopher Essex:

"When adding finitely many numbers, the order in which they are added is unimportant; any order will give the same sum. If all the numbers have a common factor, then that factor can be removed from each term and multiplied after the sum is evaluated: $ca+cb=c(a+b)$. These laws of arithmetic translate into the following linearity rule for finite sums; if $A$ and $B$ are constants, then $$\sum_{i=m}^n\left(Af(i)+Bg(i)\right)=A\sum_{i=m}^n f(i)+B\sum_{i=m}^n g(i)$$
Both of the sums $\sum^{m+n}_{j=m} f(j)$ and $\sum^{n}_{i=0} f(i+m)$ have the same expansion, namely $f(m)+f(m+1)+\cdots+f(m+n)$. Therefore the two sums are equal, $$\sum_{j=m}^{m+n} f(i)=\sum_{i=0}^{n} f(i+m)$$.
This equality can also be derived by substituting $i+m$ for $j$ everywhere $j$ appears on the left side, noting that $i+m$ reduces to $i=0$, and $i+m=m+n$ reduces to $i=n$. It is often convenient to make such a change of index in a summation."

And here the erratic questions:

1- Which are the laws of arithmetic? A quick search in the Internet is been confusing. Wikipedia is not concrete and doesn't give examples.
2- What stands for linearity in this quote?
3- Are there any linearity rules for finite sums not mentioned in the citation?

Regards, best wishes.

Edited. Reason: To complete the quote

 

[BvU]

∑i=mn(Af(i)+Bg(i))=A∑i=mnf(i)+B∑i=mng(i)
Both of the sums ∑j=mm+nf(j) and ∑i=0nf(i+m) etc...

This is pretty hard to read. Can you $\TeX$ it ?

$\$

 

[Frabjous]

Some laws for arithmetic are associative, distributive, commutative and the existence and properties of identities/inverses for addition and multiplication.

 

[malawi_glenn]

The function f is linear if and only if f(x+y) = f(x) + f(y) and if f(ax) = af(x)

 

[topsquark]

The function f is linear if and only if f(x+y) = f(x) + f(y) and if f(ax) = af(x)

True, but I would make sure to reference this to the summation operation, specifically, rather than use f(x) as in the OP. It might be confusing.

-Dan

 

[malawi_glenn]

True, but I would make sure to reference this to the summation operation, specifically, rather than use f(x) as in the OP. It might be confusing.

-Dan

Feel free to do so

 

[pasmith]

1- Which are the laws of arithmetic? A quick search in the Internet is been confusing. Wikipedia is not concrete and doesn't give examples.

The relevant rules here are that addition is commutative $$a + b = b + a$$ and associative $$a + b + c = (a + b) + c = a + (b + c)$$ and that multiplication is distributive over addition $$a(b+ c) = ab + ac$$.

Formally, the real numbers form a field, so under addition they are a commutative group with identity 0 and under multiplication the reals except 0 are a commutative group with identity 1, with multiplication being distributive over addition. Being a group carries with it the requirements that the operation concerned is associative, that there exists a unique identity element, and that each element has a unique inverse. The reals are also ordered in a manner consistent with these operations, so that if $a < b$ then $a + c < b + c$ and if $c > 0$ then $ac < bc$ but if $c < 0$ then $ac > bc$.

2- What stands for linearity in this quote?

A map $S$ between real vector spaces $V$ and $W$ is linear iff $S(f) + S(g) = S(f + g)$ for every pair of vectors $f$ and $g$ and $S(af) = aS(f)$ for every real number $a$ and every vector $f$. In this case, the set of all sequences of real numbers of a given length can be turned into a vector space under the operations of elementwise addition and scalar multiplication, $$\begin{split} (f + g)_i &= f_i + g_i \\ (Af)_i &= Af_i. \end{split}$$Summing the sequence is then a linear map from this space to the real numbers (which are themselves a real vector space): $$\sum_{i=1}^n (Af + Bg)_i = \sum_{i=1}^n (Af_i + Bg_i) = A \sum_{i=1}^n f_i + B \sum_{i=1}^n g_i.$$

 

[MidgetDwarf]

To simplify further, if we are dealing with functions y=f(x) the value f spits out when x is plugged in. Since you are dealing with functions of real numbers ( I am assuming, but it also works for n tuples of reals), those f(i) are just numbers. So the usual laws of associativity, commutavity, etc hold when adding a finite number of them. Thing's get tricky when we start adding infinitely many of these numbers together. Since we are not guaranteed that associativity and commutativity hold.

After talking about this, one starts to think of infinite series and convergence

pasmith gave an excellent explanation of a linear map.

 

[mcastillo356]

Hi, PF, to better understand everything posted, I've looked for an example to be a little specific about what we discuss.
It is better to separate the thread in two steps, and then join together
(i) Given two vector spaces $V$, $W$, a linear mapping $S:\;V\to W$ is a map preserving the following conditions:
$S(v+v')=S(v)+S(v')$ (*)
$S(\lambda v)=\lambda S(v)$ (**)
(ii) Take the vector space $V$ as the sequence of number of length $n$; for example length $n=4$. Vectors would be, for instance,
$f_i=(12,3,4,5)$, that is, $f_1=12$, $f_2=3$, $f_3=4$, $f_4=5$
and these $f_i$ would be a vector space, because we can sum them, and multiply by scalars:
$f_i=(12,3,4,5)$, $g_i=(0,-1,4,3)$, $f_i+g_i=(12+0,3-1,4+4,5+3)=(12,2,8,8)$
$f_i=(12,3,4,5)$, $\lambda=2$, $2f_i=(2\cdot 12,2\cdot 3,2\cdot 4,2\cdot 5)=(24,4,16,16)$
Now consider this vector space: $W=\mathbb{R}$. Then we can consider the function:
$S\;:\;V\rightarrow{\mathbb{R}}$, $S(f_i)=f_1+f_2+f_3+f_4$, or $S(f_i)=\sum_{i=1}^n f_i$, e.g. if
$f_i=(12,3,4,5)$, then $S(f_i)=12+3+4+5=24$
This mapping is a linear map, because it fills (*) and (**) conditions.

Regards, best wishes!

 

[mcastillo356]

Corollary:
"In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorfism or in some contexts linear function is a mapping $V\rightarrow{W}$ between two vector spaces that preserves the operations of vector addition and scalar multiplication"
Quote from the article "Linear map", Wikipedia
Thanks, PF!