# What does 'linear' mean in Linear Algebra?

Why Linear Algebra named in that way? Especially, why we call it linear? What does it mean?

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I always thought it was because Linear Algebra is pretty straight forward... – user1729 Sep 8 '11 at 10:02
Now that we have answers, we need some references to back them up. Or are they just guesses? – GEdgar Sep 8 '11 at 12:26
I think that, more accurately, it should be called affine algebra, since equations describing spaces are affine equations, usually referred-to as linear equations. – gary Sep 8 '11 at 12:55
Affine functions don't satisfy f(ax)=af(x) or f(x+y)=f(x)+f(y), and hence aren't representable as matrices without artificially extending the domain, which is presumably why the field is called linear rather than affine algebra. Or have I misunderstood you? – Chris Taylor Sep 8 '11 at 13:59
Right, and that is my point; often systems of equations to be solved are described as systems of linear equations, when the equations in the system are not those of linear objects, i.e., these objects do not go through the origin. I have seen , e.g., the "system of linear equations" given by $2x+3y=5$ and $3x-5y=6$; in neither of the two equations does y depend on x linearly. – gary Sep 8 '11 at 16:11

The subject studies linear transformations between vector spaces. Hence the name. If you read up the definition of a linear transformation in Wikipedia, you will agree that the adjective linear is apt.

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Linear algebra is so named because it studies linear functions. A linear function is one for which

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

and

$$f(ax) = af(x)$$

where $x$ and $y$ are vectors and $a$ is a scalar. Roughly, this means that inputs are proportional to outputs and that the function is additive. We get the name 'linear' from the prototypical example of a linear function in one dimension: a straight line through the origin. However, linear functions can be more complex than this (or indeed, simpler: the function $f(x)=0$ for all $x$ is a linear function!

Of course, I've brushed a lot of detail under the carpet here. For example, what kind of space are $x$ and $y$ members of? (Answer: They're elements of a vector space). Do $x$ and $f(x)$ have to belong to the same space? (Answer: No). If they belong to different spaces, what does it mean to write $ax$ and $af(x)$? (Answer: you need an action by the same field on each of the vector spaces). Do the vector spaces have to be finite dimensional? (Answer: no, and in fact nearly all of the really interesting linear algebra takes place over infinite-dimensional vector spaces).

I hope that's enough to get you started.

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Great answer! If I ever have to teach Linear Algebra, I'll try to say very early something similar to your second paragraph, in order to motivate the whole bunch of definitions... – PseudoNeo Sep 8 '11 at 16:56
Are you sure about "nearly all of the really interesting linear algebra takes place over infinite-dimensional vector spaces"? – Rasmus Nov 8 '11 at 20:43
I suspect that's a matter of personal taste. – Chris Taylor Nov 8 '11 at 23:07

I think linear refers to the fact that vector spaces are not curved. For instance, the wikipedia page for linear spaces gets redirected to the page on vector spaces. So does the one at MathWorld.

From Moore in The axiomatization of linear algebra: 1875–1940 I've learned that:

• Peano used linear systems for what we now call vector spaces. This reflects the view that linear algebra is about spaces of linear algebraic relations. (p. 265, 266)
• Pincherle was the first to use the term linear space for the concept of vector space. (p. 270)
• Hahn used linear space for normed vector space. (p. 277)
• Linear transformations as an abstract concept seem to have been introduced much later by Emmy Noether. (p. 293)
• The term linear algebra was first used in the modern sense by van der Waerden although the term can be found earlier in Weyl. (p. 294)
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+1, since I don't like anonymous downvotes. – gary Sep 8 '11 at 16:55
+1 for references! – GEdgar Sep 8 '11 at 18:16

The word "linear" means "of or pertaining to a line or lines". See http://jeff560.tripod.com/l.html for some of the earliest known uses of various types of "linear" objects.

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The link given here is useful. Why downvote? – user1551 Sep 8 '11 at 18:57
@Robert this provides some historical context that's missing from my own answer - do you mind if I incorporate the link into my answer? – Chris Taylor Sep 9 '11 at 8:22