Understanding linear expressions

[Martyn Arthur]

TL;DR Summary

How can I identify linear expressions

I understand linear expressions map straight lines.
How can I identify which expressions are linear; is x/(1-x^3) isn't linear but 1/(1-x^3) is linear?
Thanks
Martyn

 

[Martyn Arthur]

I failed to mark my just sent post to receive email alerts for replies; could this be rectified please?
Thanks
Martyn

 

[pasmith]

A function $f$ between vector spaces $V$ and $W$ is linear if and only if for every vectors $u$ and $v$ and every scalar $\alpha$, $$\begin{split} f(u +v) &= f(u) + f(v) \\ f(\alpha u) &= \alpha f(u)\end{split}.$$ For the examples you give, $V = W = \mathbb{R}$ and since it is not true in general that $(1 - x^3)^{-1} + (1 - y^3)^{-1} = (1 - (x+y)^3)^{-1}$ this function is not linear. The only linear functions from $\mathbb{R}$ to itself are of the form $x \mapsto cx$ for constant $c$.

 

[WWGD]

An map L defined on an R-Module M over a ring R, if:
1)L(cm)=cL(m), for c a constant, i.e.c is in R, and M is in M
2)L(m1+m2)=L(m1)+L(m2).

You need to be more precise on just what variable your map is linear in.

Edit: Re the above, you may have an expression $\Sigma f_i(x)X^i$, that is linear in the coefficient functions/parameters, but not on the X variable.

 

[fresh_42]

TL;DR Summary: How can I identify linear expressions

I understand linear expressions map straight lines.
How can I identify which expressions are linear; is x/(1-x^3) isn't linear but 1/(1-x^3) is linear?
Thanks
Martyn

$\dfrac{1}{1-x^3}$ is as non-linear as $\dfrac{x}{1-x^3}$ is. The only linear functions in one variable are $f(x)=c\cdot x$ for some $c\in \mathbb{R}.$

Inherited from school, people sometimes call every function $f(x)=a+b\cdot x \;\;(a,b\in \mathbb{R})$ "linear" because their graphs are a straight in the $(x,y)$-plane. However, these are strictly speaking affine or affine linear functions, means shifted by a certain value, since linearity requires $f(0)=0$ (plus those @pasmith mentioned) and thus $a=0.$ If $a\neq 0$ we should speak of affine linear functions.

 

[Martyn Arthur]

Thanks but the explanation is a little more advanced for me.
My text book (Open University MST 124 Book C) discusses integration by substitution for the first example; but then goes on specifically to explain how to integrate linear functions in respect of the second function? I seek to be able to distinguish between the two.
Thanks
Martyn

 

[WWGD]

Thanks but the explanation is a little more advanced for me.
My text book (Open University MST 124 Book C) discusses integration by substitution for the first example; but then goes on specifically to explain how to integrate linear functions in respect of the second function? I seek to be able to distinguish between the two.
Thanks
Martyn

Any chance you can show us an image of the text?

 

[symbolipoint]

Maybe I miss in finding the reason for this being complicated. A linear expression is that the variable occurs as with exponent of 1. If the expression includes that variable combined with some other number and the expression is a numerator, then this expression is not linear.

My only simple-headed example may be:
for variable x and anything else be contants,
ax is linear.
ax+k is linear.
1/x, I am guessing well, is not linear because means x^(-1).
and 1/(ax) is not linear.
and 1/(ax+k) is not linear.

 

[fresh_42]

Maybe I miss in finding the reason for this being complicated. A linear expression is that the variable occurs as with exponent of 1. If the expression includes that variable combined with some other number and the expression is a numerator, then this expression is not linear.

My only simple-headed example may be:
for variable x and anything else be contants,
ax is linear.
ax+k is linear.
1/x, I am guessing well, is not linear because means x^(-1).
and 1/(ax) is not linear.
and 1/(ax+k) is not linear.

$y(x)=ax+k$ is only in schools linear. In mathematics, it is affine linear because it fails the requirement $y(0)=0$ for linear transformations, it is shifted (affine transformation) by $k.$

 

[WWGD]

Or $\frac {1}{ax} \neq a \frac {1}{x}$

 

[Martyn Arthur]

"A linear expression is that the variable occurs as with exponent of 1. " This makes it perfectly clear; thank you.
Martyn

 

[WWGD]

Linear maps may also be described through matrices.

 

[fresh_42]

"A linear expression is that the variable occurs as with exponent of 1. " This makes it perfectly clear; thank you.
Martyn

You must be careful what you call linear. Linear is often used sloppy, as in your description

the variable occurs as with exponent of 1

This is necessary but not sufficient because you cannot write $y=ax+k$ as

Linear maps may also be described through matrices.

if $k\neq 0.$ $y=ax+0=(a)\cdot x$ is linear and a description with a $1\times 1$ matrix $(a).$ $y(x)=ax+k \,(k\neq 0)$ is strictly speaking not linear since it fails $y(0)=0$ and should be called affine linear.

However, people often speak of linear even if it is not: in school $\left[y=ax+k\right]$, in physics when the dominant approximation term is meant $\left[y(x)=a_0+a_1x+o(x^2)\right]$ or a tangent space at a point $p$ $\left[\{p\}+V\right]$, in computer science if e.g. the runtime of an algorithm is $O(n)\sim an+k$, in statistics for a linear regression $\left[y=(a)x+\vec{k}\right]$. You should not call it linear in abstract algebra, linear algebra, or analytical geometry. And you should always be aware of the two different usages of linear.

 

[WWGD]

You must be careful what you call linear. Linear is often used sloppy, as in your description

This is necessary but not sufficient because you cannot write $y=ax+k$ as

if $k\neq 0.$ $y=ax+0=(a)\cdot x$ is linear and a description with a $1\times 1$ matrix $(a).$ $y(x)=ax+k \,(k\neq 0)$ is strictly speaking not linear since it fails $y(0)=0$ and should be called affine linear.

However, people often speak of linear even if it is not: in school $\left[y=ax+k\right]$, in physics when the dominant approximation term is meant $\left[y(x)=a_0+a_1x+o(x^2)\right]$ or a tangent space at a point $p$ $\left[\{p\}+V\right]$, in computer science if e.g. the runtime of an algorithm is $O(n)\sim an+k$, in statistics for a linear regression $\left[y=(a)x+\vec{k}\right]$. You should not call it linear in abstract algebra, linear algebra, or analytical geometry. And you should always be aware of the two different usages of linear.

How can you describe a map y=ax+b with a single matrix?

 

[fresh_42]

How can you describe a map y=ax+b with a single matrix?

You can't. That was my point.

 

[pasmith]

Thanks but the explanation is a little more advanced for me.
My text book (Open University MST 124 Book C) discusses integration by substitution for the first example; but then goes on specifically to explain how to integrate linear functions in respect of the second function? I seek to be able to distinguish between the two.
Thanks
Martyn

Do you mean expressions of the form $\int f(ax + b)\,dx$?