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What exactly does 'linear' mean. For example, Hilbert space is a linear space. What is the difference between a linear and a 'non-linear' space. I am starting a self-study of linear algebra so this word is extremely important, but what exactly does it mean (especially in linear algebra).

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  • $\begingroup$ "Hilbert space is linear" doesn't sound right to me; that's either outdated or physicist's terminology. A Hilbert space is a vector space (en.wikipedia.org/wiki/Vector_space), and "linear" in this context is a property of maps between vector spaces (en.wikipedia.org/wiki/Linear_map). $\endgroup$ Oct 7, 2012 at 7:50
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    $\begingroup$ Royden's text "Real Analysis" uses "linear space" to denote a vector space. $\endgroup$
    – tomcuchta
    Oct 7, 2012 at 8:04
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    $\begingroup$ @QiaochuYuan For many people the term Linear space is synonym to Vector space, one may argue it is old fashion but who is to decide that it is wrong? $\endgroup$ Oct 7, 2012 at 8:10
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    $\begingroup$ @tomcuchta There are plenty of examples, Metric Linear Spaces, Linear spaces... $\endgroup$ Oct 7, 2012 at 8:12

2 Answers 2

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Linearity is the property satisfied by a function (i.e. function $f$ is called linear) iff it satisfies the following two properties:

  1. $f(x+y)=f(x)+f(y)$

  2. $f(ax)=af(x) $ where $a$ is a constant from the vector field in consideration.

In linear algebra you'll find that all linear functions can be represented as a matrix operation and vice versa.

Also, vector space and linear space are the same thing. Just note that term linear space is somewhat archaic and not used very much in modern books. So, I'd suggest using vector space instead.

EDIT: As pointed by qiaochu, linear space can also mean http://en.wikipedia.org/wiki/Linear_space_(geometry) though I don't think you wanted this answer. I'm just including it for the sake of completeness.

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  • $\begingroup$ "Linear space" is ambiguous; it could also refer to en.wikipedia.org/wiki/Linear_space_(geometry) . $\endgroup$ Oct 7, 2012 at 7:57
  • $\begingroup$ I don't think it's too rare. I have some textbooks published post-2000 C.E. that use the term "linear space" to mean vector space. $\endgroup$ Oct 7, 2012 at 8:05
  • $\begingroup$ Thank you, excellent answer, and I read the term 'linear space' as a property for Hilbert Space in the book 'Quantum Mechanics: Concepts and Applications' which I believe is just 3 years old. $\endgroup$ Oct 7, 2012 at 9:00
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    $\begingroup$ Sometimes affine linear functions $f(x)=a\cdot x+b$ are simply called linear (because their graph is a line). The above definition would call such $f$ linear only if $b=0$. Sometimes these two definitions may be cause of ambiguity. $\endgroup$ Oct 7, 2012 at 9:07
  • $\begingroup$ Linear functions between infinite-dimensional vector spaces cannot be represented by matrices (unless you generalize to infinite matrices, which don't always behave nicely -- and even so you'll have the problem that the vector space does not always a basis that can be described explicitly). $\endgroup$ Oct 7, 2012 at 10:29
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Linearity does have a precise definition. A function $f$ is linear if it is homogenous ($\alpha f(x) = f(\alpha x)$) and additive ($f(x+y) = f(x) + f(y)$). To keep things concrete, let $\alpha$ in the definition be a real number and imagine $f$ is a map from $\mathbb{R}$ to $\mathbb{R}$. What do these conditions mean?

Well, consider the value of $f$ at $0$. Since $f(0) = f(0+0) = f(0)+ f(0) = 2f(0)$, we must have that $f(0) = 0$.

Also, for any $x\neq 0$ we have that $f(x) = xf(1)$. So, if we call $f(1) = a$, we've learned that $f(x) = ax$. That is, the 1D functions that satisfy the conditions for linearity are lines that pass through $0$. Lines in the plane are very nice objects that we know a lot about.

The power of the definitions above is that they can be generalized in at least two ways. In Linear Algebra, you'll see that for functions $f:\mathbb{R}^n\mapsto\mathbb{R}^m$, the conditions of linearity imply many of the nice properties familiar from simple one-dimensional lines in the plane. This turns out to be very useful (and is taken further as in, just for example, functional analysis).

If you go further (or maybe later on in the semester, depending on your course), you'll see we can also fruitfully drop our condition that $\alpha\in \mathbb{R}$.

Linear spaces (ie, vector spaces) and functions are very well studied and have many applications across pure and applied mathematics, as well as in the sciences. As a result, you'll sometimes hear people (across engineering and the sciences) speak somewhat more loosely about "problems" which are "linear". In this context, you'll often think of $x$ as an input to some system. A "linear system" is one in which when you scale the input by $\alpha$ (say, pedal faster on your bicycle), the output of the system scales by just the same amount (your bike moves faster a proportionate amount). That is, $f(\alpha x) = \alpha f(x)$.

A nonlinear system, then, might be one where a tiny change in input results in a huge change in output (you pedal faster and your bicycle explodes). Explosions, turbulence, and the "butterfly effect" are typical examples.

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    $\begingroup$ "Linear" has a precise definition in linear algebra. Across mathematics as a whole it less precise -- in particular, much of the time "linear" is used for things that are allowed to contain a constant term, and things without constant terms are called "homogeneous linear" systems (etc.) $\endgroup$ Oct 7, 2012 at 10:33

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