For my 3rd and (thank god) final math class I can take calculus III, linear algebra or intro to differential equations. Normally the school has descriptions of each class and what its over but I can’t find any for the math classes.
I dont think calculus is for me anymore so I’m looking at linear algebra as a potential 3rd class. What is an overview of the material covered in linear algebra? I talked to my brother the math major (he took it as a freshman) and he said it was about matrices, but I have almost no experience with those. I had some in Inorganic chemistry but not much. What else is there to this branch of math, and what is its purpose and what is it based on?
I haven’t had a chance to read all of that since I just found it but it is a linear algebra text.
For that matter, what are differential equations? I assume calc III is just more differentation/integration but with 2 variables instead of 1 but I don’t know what to expect in a class like Linear Algebra or Diff. equations
Linear algebra deals with multidimensional vectors, state equations, and the like. A lot of matrix math. I deal with it in controls work and navigation; can’t think of something a chemistry major would use it for, but then I’m an electrical engineer.
The word around IUPUI is that MATH 511 (graduate level linear algebra) is pretty easy. It’s a required class for grad students in ECE.
On the other hand, I also use DiffEq quite a bit, and it’s not bad if you’e mastered calculus. My guess wuld be that it’s more relevat to a chem major.
Calc three will be differntials and integrals with (like you said) two variables, but it will quickly move on to three, four etc… If you liked math and calc, go for it. Calc three was one of my favorite classes (a tossup between that and geometry). Diffeq, IIRC wasn’t that difficult, just a lot of writting, as in equations that can take a page or more, and god forbid you screw something up at the beginning. If you can pay attention and are okay with not always understanding but just going through the motions, it shouldn’t be too bad. If you’ve done implicit differentiation in calc already, that’s what the class is about. (Here’s a tip though, if you have bad hand writting, put a line through your Z’s turing a Z into a 2 is a tuff mistake to find and correct.) Your best bet is to talk to your counsler or someone in the math department and ask them for their recommendation, they’ll be the best judge of what will be best for you in your specific situation.
If you don’t like math that much, don’t take differential equations. Unlike calculus and linear algebra, you don’t have a set procedure for solving problems. Rather you have to learn how to solve different specific classes of problem, and remember what to do in each case.
Linear algebra is fun, in my opinion. Mostly manipulating matrices (groups of numbers) – it’s like solving a Rubic’s cube, but with rows of numbers. There are some definitions to memorize, but I think it’s all fairly intuitive. Calculus III will be a lot like your previous calculus classes – tacking on more dimensions doesn’t change the basic concepts, so that might be easiest, although if you’re sick of calculus you may not want more of the same.
I remember reading that you didn’t like calculus. I’m not sure if you would like linear algebra. It would depend on what kind of class it is. I think there are two kinds of linear algebra classes. (At my school, there was “matrix algebra” and “linear algebra”.) One kind is geared more towards engineers, and involves more computation. The other kind is geared more towards math majors, and probably involves proofs. They both teach the same basic theory, but the emphasis is different. I think you should find out what kind of class it is before taking it. If it is a proof-based class, and you are not familiar with mathematical proofs, you might be in trouble.
Another thing to consider is that many people find calc III to be easier than calc II, for some reason.
Anyway, I will try to provide you some idea of what linear algebra is. Suppose you want to solve the following system equations for x_1 and x_2.
ax_1 + bx_2 = y_1
cx_1 + dx_2 = y_2
You can also write this in matrix form. I can’t quite draw matrices here, so I will have to describe it.
Let A be the matrix with [a b] in the first row and [c d] in the second row.
Let x be the column vector with x_1 in the first row and x_2 in the second row.
Let y be the column vector with y_1 in the first row and y_2 in the second row.
Then, you can write the above equations as Ax = y. Matrix notation allows you to write systems of linear equations in a more compact form. A matrix is actually a representation of something called a linear mapping with respect to particular bases. Multiplication of matrices corresponds to composition of linear mappings.
As for differential equations, you will basically be solving problems like the following:
Anyway, everyone will tell you that linear algebra is the study of matrices, but that just ain’t so. It’s the study of linear maps between vector spaces, and you can use matrices to do that.
A vector space is a pair of sets V and F where you can add together elements of V and multiply them by elements of F, and those operations satisfy some basic rules. A linear transformation is a function T from one vector space to another which satisfies T(au + bv) = aT(u) + bT(v). Shortly after the notion was formalized, it was discovered that every such map can be represented as a matrix.
Linear algebra will involve far less plugging and chugging, and more proofs. It’s a good class to take, because the proofs tend to be fairly simple, and the deductive reasoning skills you develop will serve you well.
A typical linear algebra course might start out with systems of linear equations, which you probably first encountered in a high school, or possibly college, algebra class. The linear algebra course would go into ways of solving systems using matrices, and interpretations of systems of equations in terms of vectors, and theoretical stuff behind what kinds of systems/vector equations can have what kind of solution sets. You’d study matrices and their properties; vector spaces; and linear transformations (which can be described using vectors)—all from a more or less theoretical standpoint depending on the particular class you’re taking—see bitwise’s post. Personally, I enjoyed the linear algebra classes I took and found them to be about the easiest of all my math classes.
Note that linear algebra doesn’t really have anything to do with calculus, except that occasionally concepts from calculus are used as examples in linear algebra. And vectors are big in both Calculus III and Linear Algebra, but in different contexts.
As for Differential Equations, ideally, you would take DiffEq after taking both Calculus III and Linear Algebra, though you’d have to check the pre-reqs at your school to see if it was absolutely necessary. But some of the things that come up in DiffEq (including partial derivatives, linear independence, and using matrices to represent systems of equations) make more sense if you’ve seen them before in Calc III and/or Linear Algebra. And a good Diff Eq class might well make use of just about everything you’ve learned in math up until that point.
I think of linear algebra as the formal link between matrix calculus and linear transformations. In a rigorous linear algebra course, everything you’ve learned involving matrices will be a special case of a more general theory involving vector spaces, linear transformations, polynomials, determinants, decompositions, and more. Not a simple subject by any measure.
