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

Eonil
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    I always thought it was because Linear Algebra is pretty straight forward... – user1729 Sep 08 '11 at 10:02
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    Now that we have answers, we need some references to back them up. Or are they just guesses? – GEdgar Sep 08 '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 08 '11 at 12:55
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    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 08 '11 at 13:59
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    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 08 '11 at 16:11
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    @GEdgar, I've added some references in my answer. – lhf Sep 08 '11 at 16:53
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    in LP, the objective function and the constraints are all of degree 1 (linear). Also, linear algebra is used in solving the problems. The above derives one to think that this may be part of the reason for the name. Who came up with the name anyway? – NoChance Feb 03 '18 at 06:50
  • However it depends what is linear in what. For example a linear change of coordinates : given $(a,b)=u(c,d)+v(e,f)$ then $u$ and $v$ are not linear in the vectors $(c,d),(e,f)$ – QuantumPotatoïd Dec 24 '22 at 06:34
  • Except if the inverse of the matrix of the basis vectors is linear in the latter matrix (in particular unitary matrices satisfy this) – QuantumPotatoïd Dec 28 '22 at 19:59

5 Answers5

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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 a lot of really interesting linear algebra takes place over infinite-dimensional vector spaces).

I hope that's enough to get you started.

Chris Taylor
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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 08 '11 at 16:56
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    Are you sure about "nearly all of the really interesting linear algebra takes place over infinite-dimensional vector spaces"? – Rasmus Nov 08 '11 at 20:43
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    I suspect that's a matter of personal taste. – Chris Taylor Nov 08 '11 at 23:07
  • Good answer, but I might add that prototypically, we usually see examples of linear functions in two dimensions: y = mx +b, where x and y are dimensions of a coordinate plane. – spacedustpi Mar 03 '20 at 02:13
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    I would add that linear transformation maps line to line. We can interpret geometry as the study of some class of transformations (of a space) that preserve particular properties. I.e. we can talk about "linear algebra and linear geometry" as a study of the transformations that maps line to line ("preserve linearity"). – Aleksander Gurin Dec 23 '23 at 12:53
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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)
lhf
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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.

Robert Israel
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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.

George
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Linear algebra is concerned with linear functions and linear equations. They are used to find the points that make a up a line. Linear algebra is simply the algebra concerning lines. Linear means along a straight line.

Geizio
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