# I found this. What is it?

by clanijos
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 P: 27 So I found this on the ground before I took my ACT this morning: http://imgur.com/a/f9pu0 It doesn't appear to be very difficult, but I don't really know what it is. It's probably pretty simple, I just don't recognize it. If anyone could tell me what this represents, or what type of classes it might be associated with, that would be great! (Also: If you're Josh Carison, I don't know who you are, but I think I may have found your homework)
 Mentor P: 7,318 That is linear algebra.
 P: 27 So, It really is just vectors and matrices? I thought maybe there was something else to it. Hrm. Ok then.
 P: 150 I found this. What is it? While vectors and matrices are an important part of linear algebra, they are far from representative of the whole subject. Linear algebra deals primarily with vector spaces, which are sets with addition and scalar multiplication satisfying certain axioms (regular vectors in Euclidean space also fall under this definition). A function T:V→W (where V and W are vector spaces) is called a linear transformation if T(u+v)=T(u)+T(v) and T(cv)=cT(v) for all u and v in V and all scalars c. With the choice of a basis for a vector space V (a set of vectors which acts like a "coordinate system"), the elements of V can be written as lists and linear transformations from V to itself can be written as (square) matrices. Thus matrices are important in linear algebra, but ultimately just represent linear transformations in a particular basis (which makes calculations more convenient).
 P: 27 That's quite helpful! Thanks!
 P: 150 I don't like people to fall into the "matrix trap" (to get the impression that linear algebra is nothing more than solving systems of linear equations with matrices, etc). Matrices don't do the subject justice.
 P: 27 Since the original purpose of this thread has been fulfilled, I think it's ok to go ahead and go off-topic for a bit. I'm pretty sure Keanu Reeves is in the "Matrix Trap".
P: 906
 Quote by A. Bahat I don't like people to fall into the "matrix trap" (to get the impression that linear algebra is nothing more than solving systems of linear equations with matrices, etc). Matrices don't do the subject justice.
indeed, matrices stop being a useful way to think about linear transformations when we have non-finite bases.

still, it IS profitable to think of Rn with respect to the standard basis, and using standard bases for Rn,Rm allows us to easily compute the values of a linear transformation T:Rn→Rm using an mxn matrix.

and studying matrices does give one a "feel" for the ways in which an "abstract" linear transformation behaves. and often, in physical applications, one has certain "coordinate systems" in mind, from the outset, and one is looking for some numerical quantity which expresses something we are measuring in these coordinate systems (like, say, a force vector).

p.s.: i don't think the use of "matrix" in the famed movie series has anything to do with linear algebra, but rather is the broader english meaning of "an inter-connected web"

(matrix: < L. matrix, "expectant mother" < L. mater, "mother"....later "origin/source" or "place of development", and then (c. 17-th century) "embedding/enclosing mass").
 P: 27 Yep.
 P: 117 By the way if you ever run into Josh Carison you could tell him that in addition to using very very sloppy notation he made some errors in his homework.
 P: 27 Yeah, I assume that's why he cast it so hastily aside! Thanks for the discussion everyone!
 P: 429 Matrices for doing, linear maps for understanding.
 P: 150 Well said.
 P: 27 Indeed. Good work out of you.

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