Why do we call y = mx + b linear (in high school)? Why don't we call it affine?
and linear what is actually linear
$$f(x+y)=f(x)+f(y)$$ $$f(ax)=af(x)$$
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2Why do you think it's called "linear"? Maybe because the graph is a line? Mathematics is not identical with pedantry, it never was very strong at consistent terminology. Just make clear what you are talking about in any given context. – Dec 17 '17 at 06:33
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I just feel like students get the wrong impression when they hear linear algebra especially since y = mx + b is algebra and linear – user29418 Dec 17 '17 at 06:35
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Maybe you could use some reading of Paul Lockhart’s A mathematician’s lament. – Jonatan B. Bastos Dec 17 '17 at 06:38
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Almost all of linear algebra (and linear programming) is about affine functions. You won't change that, and anybody trying would just look funny. – Dec 17 '17 at 06:40
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The right question is why do we abandon the perfectly natural and intuitive definition of "linear" in high school and later change it to insist the dang things send $0$ to $0$. – zhw. Dec 17 '17 at 07:10
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As written in den comments, you have to consider the context. There are also other definitions which are not consistent in mathematics. An other example is the set of natural numbers. Some authors say that $0$ is a natural number but other say that the natural numbers start with $1$.
In high school you call $f:\mathbb R\to\mathbb R$, $f(x)=mx+b$ a linear function and sometime we still use it in analysis for example like this:
Define $f:\mathbb R\to\mathbb R$ continuously by $$ f(x)=\begin{cases}0 & x\leq n\\1 & x\geq n+1\\linear & x\in[n,n+1] \end{cases}. $$ Here obviously $f$ is not linear in sense of linear algebra on $[n,n+1]$, but everyone knows how to understand it.
On the other hand, if you consider vector spaces $V$ and $W$ and a linear function $F:V\to W$ then it is obvious, that $F$ has to be linear in sense of linear algebra. So from the context there is no misunderstanding.
Mundron Schmidt
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