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A recent question I asked got me thinking.

The map $T:\mathbb{R} \to \mathbb{R}$ defined $Tx = x-1$ is not linear. But it seems strange to me that it isn't. Compare with the graph $y=x-1.$

I shudder to think how many times I've thought of such maps as linear when they're not. It's counterintuitive??

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  • $\begingroup$ Is $T(1+2)=T(1)+T(2)$? Is $T(3\cdot 1)=3T(1)$? It is true that an equation like $x-1=0$ is called a linear equation. Perhaps a slight inconsistency of terminology. $\endgroup$ – André Nicolas Sep 4 '13 at 18:29
  • $\begingroup$ @AndréNicolas You misunderstand my question. I know it's nonlinear but it looks like it should be linear. $\endgroup$ – BigUser Sep 4 '13 at 18:30
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    $\begingroup$ The seond part of my comment kind of addresses why you are uncomfortable. The term linear is used in two different senses. $\endgroup$ – André Nicolas Sep 4 '13 at 18:32
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This is where you start wanting to get comfortable with the difference between linear and affine... an affine map is basically a shifted linear map. Something like $A(x):=T(x)+b$ where $T$ is linear.

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$T(x)=x-1$ is a linear polynomial but only an affine transformation. The adjective "linear" has subtly different meaning in different branches of math. (Now go find out what different meanings "normal" can have in different branches of math)

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From my experience, maps as in your example are often called linear in school but affine in university, where the term linear is reserved to maps respecting sums and scalar multiples of vectors, see here.

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The counterintuitive thing about the term "linear" is that there are many line-like functions that are not "linear" in the sense of a linear transformation. If you want to extend your definition to include these line-like transformations, you want to look into affine transformations.

The line $y=x-1$ is, however, a "linear equation", since we can write this set as $y-x=1$. That is, if we take $T:\mathbb R^2\to \mathbb R$ given by $T(x,y)=y-x$, the equation $T(x,y)=1$ has that graph.

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Introductory textbooks, which usually deal with real-valued functions on the reals, tend to refer to linear functions as those whose graphs are (non-vertical) lines in the plane. This is fairly natural, certainly, and it seems just as natural to generalize this idea to vector spaces, so that linear maps are functions $$T:\Bbb F^n\to\Bbb F^m$$ (where $\Bbb F$ is some field) given by $$T(x)=Ax+b,$$ where $x$ is an arbitrary $\Bbb F^n$ vector, $b$ some fixed $\Bbb F^m$ vector, and $A$ some fixed $m\times n$ matrix with entries in $\Bbb F$.

Unfortunately, linear maps are usually defined (in more advanced texts) with the requirement that $T(0)=0,$ so we need $b=0$ in the above definition in this case. The more general functions $T(x)=Ax+b$ are known as affine maps.

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