Algebra 2 questions

[Pohatu: Uniter of Stone]

I was recently learning matrix addition/subtraction in Algebra 2, and then we started learning about matrix multiplication. Immediately, my brain exploded. It made no sense. It makes sense now, but I still have a question- Why does matrix multiplication work the way it does, instead of straight across like matrix multiplication? I tried searching google, but all that popped up for me were math help websites that explained how it worked but not why. If anyone is good with matrices here, could you answer my question: why does matrix multiplication work in such a weird way, and why can't you just multiply the numbers in the same locations like you would when you add or subtract?

 

 

Thanks.

[LockmanCapulet]

I'm currently taking linear algebra, which deals a lot with matrices, but I unfortunately do not have an answer for you. I guess it's just one of those things that probably has an explanation that is really long and complicated and doesn't need to be understood to perform the operation. The answers I find when I search for it talk about linear transformations; my best guess is it has to do with the fact that matrices can be said to be composed of vectors, but I have no idea what that would imply. I would do more reading about it if only I could understand the current lessons! ^^' Sorry I couldn't be of more help.

[~kh]

It's a concept that mostly has application in computer science. It's basically an array, except much more conceptual than that. They are very indeed very confusing. I'm kind of surprised that you're leaning about this in Algebra II; I went all the way to Discrete Mathematics before I came across them.

[LockmanCapulet]

I just learned that the next unit in my linear algebra class is on linear transformations; hopefully soon I may have some semblance of an answer for you!

 

Also, I agree with Kahi, I'm surprised you're learning this in Algebra II. I didn't touch this stuff 'till Multivariable Calculus.

[Bfahome]

I remember doing some stuff with matrices in high school. I don't remember what exactly (I didn't understand much of it at the time), but it was mostly surface-level stuff (like you said, adding, multiplying, etc.). I got into more complex matrix math when learning linear algebra in college. I guess this is more directed at the surprised reactions to your post than to your post, but I don't really have an answer either. Matrices can get pretty complex with how they work, and while I have a decent understanding of them I don't think I know enough to be able to actually explain it.

 

Your best bet would probably be to ask your teacher. If they don't have an answer, mabe they can direct you to a resource that does. If you can't find anyone or anything that helps, then hold on to the question so you can ask future teachers or professors if you still want or need to know.

[Space: Ocean of Awe]

Matrices are weird. There's totally a reason behind their weirdness, but it's been so long since I've done anything with them, I don't even know if I ever knew the reason. A fascinating question, though, and I'll see if I can find anything out about it. You've made me all curious now!

 

=)

[Axilus Prime]

Argh, I'm dealing with matrices in Algebra 2 now also. And you're actually a step ahead of me, I can't solve anything involving them without my notes on the exact process for how to do it right next to me.

 

Even then I'm prone to making a mistake in one place or another and then having to restart from there.

 

But then, I've always had trouble with mathematical concepts that can't work with physical objects. Geometry, I thought was hard, but now that I'm in Algebra 2 I realize how much more sense it made.

[Mysterious Minifig]

I took a look at my linear algebra text book and it kind of had an explanation. I'm attaching the two images that kind of show the proof. I'll try to explain some of the stuff in there since I'm not sure how much of the concepts your class has covered or if you know all of the notation.

 

Image 1

 

The first thing is the definition of a Matrix multiplied by a vector. I'm assuming you understand most of he definition. It's basically how we switch between a system of linear equations and a matrix. Where is says x is in Rⁿ, that means the dimension of the vector x, or the number of entries in x. The bottom half is somewhat of a proof for multiplying two matrices.  One note on that is that b₁ - b_n are column vectors like x (without the sub script) while x₁ - x_n are constants.

 

If you don't really like letters, I redid it it using numbers to show what all the letters are saying. I guess I didn't completely finish, but you should be able to tell that it works out to match the rules for matrix multiplication.

 

Image 2

 

Please let me know if you need additional clarification on any of this.

 

Edited October 31, 2014 by Mysterious Minifig

[Taka Nuvia]

Trying my hand at a small explanation, but I heavily suggest taking these things with a grain of salt, as I'm not an expert in these matters myself though I can work with matrices just fine.

 

The shortest explanation I can think of is that matrices often depict some kind of transformation; there are, for example, rotation matrices that contain several sinus- and cosinus-terms, and if these matrices operate on a vector, the vector gets rotated by a certain angle

 

Multiplying matrices can be seen as a composition of their operations - say, rotating around the z-axis for a certain angle and then around the x-axis for another angle. Which is also why the matrix product isn't necessarily commutative.

 

Quick sketch:

In both cases we get the vector r_i'' by rotating our original vector r around the z-axis (angle theta) as well as the x-axis (angle phi). As you can see the order does make a difference. So if the rotations around the axes were described by the matrices Z and X, you would find that

 

Z * X * r

 

is not the same as

 

X * Z * r

 

A 'normal' vector product is commutative, because it does not matter whether you have ( x₁ - x₂ ) or ( -x₂ + x₁ ) for the first component. So apparently we have two different 'processes' at work, which is also why we require different methods of multiplying.

 

At least that's the way I understand it; might be a bit incorrect in a few places. Hopefully not too confusing. D:

 

~~

 

[...] The answers I find when I search for it talk about linear transformations; my best guess is it has to do with the fact that matrices can be said to be composed of vectors, but I have no idea what that would imply. [...]

 

One way of describing a linear transformation is using an array that expresses how the base vectors (unit vectors?) change under transformation. These changes are then put into the matrix, which, if operating upon a vector, will transform its components accordingly.

 

Edit: removed a few bits that were certainly wrong, or at least odd. Sorry 'bout that. :/

Edited November 3, 2014 by Fairy Paladin

[Dallior]

Yeah, I'm Algebra IIing  (in the 8th grade ) and I haven't seen anything close to what youre describing.

[Jacks]

By all rights I should've failed Algebra 2. The only reason I didn't was because I had one of those teachers who, when he saw how many of his students were failing, bumped everyone's grades up so that the school wouldn't reevaluate him So the only reason I'm posting here is to mention that there's totally a way to do basically anything you need to do to a matrix on a TI-83, or whatever y'all are using these days, without having to think about how or why if you don't want to worry about that

 

(come to think of it, I don't remember if we actually learned matrices in algebra 2, or if it was only a thing in precalc, which I also should've failed. anyway)