What does 'linear' mean in Linear Algebra?
Why Linear Algebra named in that way?
Especially, why we call it linear? What does it mean?
Solution 1:
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.
Solution 2:
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)