Why Linear Algebra named in that way?
Especially, why we call it linear
? What does it mean?
What does 'linear' mean in Linear Algebra?
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14I always thought it was because Linear Algebra is pretty straight forward... – 2011-09-08
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2Now that we have answers, we need some references to back them up. Or are they just guesses? – 2011-09-08
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0I think that, more accurately, it should be called _affine_ algebra, since equations describing spaces are affine equations, usually referred-to as linear equations. – 2011-09-08
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3Affine 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? – 2011-09-08
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2Right, 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. – 2011-09-08
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1@GEdgar, I've added some references in my answer. – 2011-09-08
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1in 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? – 2018-02-03
5 Answers
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.
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1Great 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... – 2011-09-08
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1Are you sure about "nearly all of the really interesting linear algebra takes place over infinite-dimensional vector spaces"? – 2011-11-08
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1I suspect that's a matter of personal taste. – 2011-11-08
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)
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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1The link given here is useful. Why downvote? – 2011-09-08
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0@Robert this provides some historical context that's missing from my own answer - do you mind if I incorporate the link into my answer? – 2011-09-09
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.
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.
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1The question is more than 6 years old and have several answers that say mostly the same, don't waste your time like that. – 2018-03-25