Linear Algebra Exam: Defining Vector Spaces Vocab

[Sting]

I have an upcoming Linear Algebra exam and my textbooks are really vague in defining certain concepts (and he didn't limit the ambiguous nature to Linear Algebra. His Calculus book is the same way).

Would someone mind helping me define or determine Determination tests for concepts like span, subspace, basis, rank, dimension, row space, column space, linear combination, etc.?

Thanks

 

[ahrkron]

I think it will be better if you post a couple specific questions.

 

[Sting]

Okay. It's really just a request for less vague definitions.

I figured out the concept of span (but it took a while).

However as for a specific question, how does one determine the row space and column space?

 

[damgo]

You can find rigorous definitions at http://mathworld.wolfram.com (but there are often multiple equivalent ways to define something, so these may slightly differ from your book's.)

IMHO you should start by reading and figuring out the exact definitions of a vector space and of a linear transform. Once you understand those, the other defs are easy.

 

[Sting]

Well, the test was last Wednesday but I found the Schaum's Outlines for Linear Algebra and around four really good textbooks so I purchased them and studied throughout the break.

I basically understand the concepts (the nature or neccessity is another story) so about the exam, I think I did decent (but I heard there was a curve so I just might get an A or a B (I'm sure it's at least a B without the curve).

Thanks Damgo

 

[Mulder]

Originally posted by Sting
I basically understand the concepts (the nature or neccessity is another story)

Hehe I'm currently learning very similar stuff (all you mentioned in first post) and am also yet to see either nature or necessity

 

[Sting]

Hehe I'm currently learning very similar stuff (all you mentioned in first post) and am also yet to see either nature or necessity

lol, I don't think there is any practical applications of it.

Mathematicians just sat together one day to smoke some pot and BANG! Linear Algebra is born...

 

[bogdan]

Linear Algebra -> matrices -> tensors -> SR,GR,deformations,fluids...physics...

 

[enigma]

Electronics... control systems... computer graphics... deformations analysis...

 

[Ben-CS]

3-D video games and computer games use a lot of linear algebra. Bucketloads of it. So much, that game designers have developed or applied advanced compact techniques of representing linear transforms.

 

[damgo]

My first "moment of clarity" in linalg was when I saw why matrices and linear transforms were the same thing... work on figuring this out. :)

Real 3-D space -- R3 -- is a happy vector space, and lots of the things we want to with, like rotations or scalings, are all linear transforms. Computers do incredibly large amounts of matrix manipulations, all the time.

Plus about all of physics and lots of math is built on top of linear algebra... partially because as the old saw goes "everything is linear to first order." :) It seems weird at first, but its importance will soon become apparent... I hardly even think about it as math anymore because it's such a built-in part of the way I think...

 

[quantumdude]

I took linear algebra my freshman year, and did not use much of it until my first year of grad school, in Quantum Mechanics I. Others were in a similar boat, and the professor found that he had to give us a refresher on the vector space axioms.

 

[Sting]

Guys, I was joking when I had mentioned the lack of practical applications in Linear Algebra.

However, I have yet to see an actual application of it since a good bit of it has been theoretical.

Oh, before I forget, I got a 96 on my Linear Algebra exam! Second highest test score in the class so that's a bit of relief.

I took linear algebra my freshman year, and did not use much of it until my first year of grad school, in Quantum Mechanics I.

Dr. Postell (my Linear Algebra professor): "...this is quite a common inner product. Physics majors <looks at me> will use this extensively..."

Tom, would you mind showing us an example of how Linear Algebra comes into place with QM?

Plus about all of physics and lots of math is built on top of linear algebra... partially because as the old saw goes "everything is linear to first order." :) It seems weird at first, but its importance will soon become apparent... I hardly even think about it as math anymore because it's such a built-in part of the way I think...

Wow, that's cool. Want to hear something ridiculuous? Linear Algebra isn't a requirement for Physics majors at my school. In fact, I could drop the class and take a computer class and get my requirement filled for that area. I'm taking Linear Algebra out of concern because I know it'll come into place in Physics.

 

[RuroumiKenshin]

what's linear algebra?

 

[Sting]

Linear Algebra is the study of matrices and the properties of matrices.

And Majinvegeta, I haven't forgot about my linear algebra notes to send to you. My Microsoft Word has developed a virus on it so sending notes is very difficult but I will send my notes as soon as I can.

 

[Mulder]

Hey Sting did you decide on which major you would take - Maths/phys or both I can't remember whether you did or not..

 

[Sting]

Hey Mulder,

Actually, Physics definitely, but I want to double major in either Physics and Mathematics or Physics and Electrical Engineering.

The EE is for job security and finances but the math degree is for temporary job security, fun, and the potential opportunity for working in genetics (last time I read, they need statisticians).

So, techinically, I'm still trying to look over all my options.

Wish me luck!

 

[RuroumiKenshin]

groovy! I can't wait.

What is it used for? A lot of math(i'm thinking of fractals) doesn't apply to the physical world. So what's its use, how does it help us improve things?

 

[Hurkyl]

The main importance, IMHO, is that Linear Algebra is a language, it gives us the descriptive power to talk about things that would otherwise be either very cumbersome or very vague.


For instance, one of the things matrices give us is the power to talk about a system of equations as a single unit, and do a variety of operations that make sense to do on the system as a whole.


Another thing it gives us is the power to speak rigorously about some geometric concepts. For example, consider the surface given by the parametric equations:

x(s, t) = s
y(s, t) = s^2 * t
with -1 < s < 1 and -1 < t < 1

If you plot this suface, you will notice that it's kinda sort of pinched into a 1-D surface at the origin. We can prove this rigorously as follows:

First, compute the jacobian of the above equations. The jacobian of a system of functions is simply the matrix who's i-th row and j-th column contains the derivative of the i-th function with respect to the j-th variable. In this case, the jacobian of the transformation is:

[1, 0]
[2st, s^2]

If you plug in 0 for s, you get:

[1, 0]
[0, 0]

Which is a rank one matrix. (The rank of a matrix is the the dimension of its row space; i.e. the number of linearly independent row vectors in the matrix) Our vague idea of a 1-dimensional surface coincides exactly with jacobians that have rank 1! Similarly, if a system of equations describes a 2-d surface the matrix has rank 2. For example, at the point s = 0.5 t = 0.5 the jacobian of the above system is:

[1, 0]
[0.5, 0.25]

Which is rank 2, as we'd expect from plotting the surface.

Hurkyl

P.S. bleh just saw where you said you're joking
P.P.S can you do superscripts in 3.0?

 

[damgo]

Linear Algebra isn't a requirement for Physics majors at my school.

<gapes in disbelief> Congrats on your test tho!

Linear Algebra is the study of matrices and the properties of matrices.

<smashes head repeatedly against the wall>

Hurkyl hit the nail on the head... trying to do QM without linear algebra is like trying to write an essay without sentences. For example, in the general formulation of QM:

A state of a system is represented by a vector |[psi]> in a special type of complex vector space called a Hilbert space. Observable properties are represented by a special type of linear transformation called a Hermitian operator. (you will probably see symmetric matrices, which are very similar.)

The average value of some observable A in state |[psi]> is then the inner product of A([psi]) and A, written <[psi]|A|[psi]>. And so on... in general the inner product tells you how much "overlap" ther is between two states. It takes a while to grasp how it all fits together, but it's so pretty when it does...

 

[bogdan]

fractals->biology...physics...and a lot more...

 

[enigma]

Originally posted by MajinVegeta
groovy! I can't wait.

What is it used for? A lot of math(i'm thinking of fractals) doesn't apply to the physical world. So what's its use, how does it help us improve things?

I'm taking a class in control systems right now. All that class is, is fun with imaginary numbers.

The class is blowing me away. The results are really neat, but the math is crazy.

Me in class--->

 

[Sting]

<smashes head repeatedly against the wall>

Was there something inaccurate about the definition I gave?

<gapes in disbelief> Congrats on your test tho!

Thanks! But knowing that it isn't even a requirement is hard to believe isn't it?


Thanks Hurkyl and Damgo for the explanations. It looks mindboggling but I know with a little patience and hardwork...

 

[damgo]

No, I know that's how they teach it... and it's sort of accurate, but kind of like describing high school geometry as "the study of two-column proofs." Linear transforms and linear spaces are the objects... matrices are just a particular way of writing them down, albeit a very useful one. :)

 

[rutwig]

Originally posted by Sting
Linear Algebra is the study of matrices and the properties of matrices.

Not exactly. It is the structural study of vector spaces. A matrix corresponds to faithful representations of vector spaces or to their visualization as endomorphisms, i.e., as mappings between vector spaces.
Concerning the list: span, subspace, basis, rank, dimension, row space, column space, linear combination

span: it corresponds to the subspace generated by a family of vectors

subspace: a subset of a VS which also satisfies the axioms

basis: a maximal set of linearly independent vectors

rank: in matricial language, it is the dimension of the image space by an endomorphism

row/column space: this depends whether you use the row or column notation to denote vectors. Strictly speaking, the row vector associated to a column vector is an element of the dual space (also linear forms space)

linear comb: a sum of vectors which scalar coefiicients

 

[selfAdjoint]

A matrix is what you get when you put a linear transformation into a particular coordinate system. As long as you stay at the LT level you can discuss and prove theorems without ever using a coordinate system, and many people think that that is better.

But in my opinion a lot of the confusion of beginners in Hilbert spaces and states and operators would be reduced if these things were presented as row vectors, column vectors, and matrices.

 

[Hurkyl]

Of course, doubly infinite matricies can lead to a whole new can of worms.

Hurkyl