Subspaces, R^n How to visualize?

[bobby13]

Hey,

I have no problems dealing with vectors in space, R^3. But I am having a lot of trouble with vectors in R^n. One of my basic questions is what is R^n. I mean doesn't the vector space already encompass everything? How do I visualize R^n vectors? Can you recommend any good online tutorials that explain how to approach problems in the subspace units?

Any help would be great!

Thanks!

 

[matt grime]

Don't visualize it. Period. These are not spaces that have any physically visible manifestation really, so don't bother - it is unnecssary to do so, and unhelpful in the long run if you only attempt mathematics that is 'visualizable'.

R^n is just the set of n-tuples of numbers, and that is all you need.

 

[JasonRox]

Don't visualize it. Period. These are not spaces that have any physically visible manifestation really, so don't bother - it is unnecssary to do so, and unhelpful in the long run if you only attempt mathematics that is 'visualizable'.

R^n is just the set of n-tuples of numbers, and that is all you need.

I can't even visualize R^3 half the time.

Look at the bright side of R^n, you'll never be asked to draw a graph.

 

[Hurkyl]

I dunno, I've seen people draw diagrams in much uglier spaces than than R^4. That's where the real fun begins.

 

[bobby13]

Thnx! I think I am starting to get the hang of this stuff, after constantly and constantly reading over notes and trying examples. Quick question. In one example it says,

Let U = { X in R^n | AX = 0 (Zero Vector)}
Is U a subspace?

Am I correct in reading the question as, X is in R^n if and only if AX = 0. That is, when checking all three conditions, I should get the zero vector?

 

[hypermorphism]

Thnx! I think I am starting to get the hang of this stuff, after constantly and constantly reading over notes and trying examples. Quick question. In one example it says,
Let U = { X in R^n | AX = 0 (Zero Vector)}
Is U a subspace?
Am I correct in reading the question as, X is in R^n if and only if AX = 0. That is, when checking all three conditions, I should get the zero vector?

It is the definition of a set U. It says X is in U iff X is a vector in R^n such that AX=0. Now you have to determine whether U satisfies the axioms of a vector space.

 

[matt grime]

Thnx! I think I am starting to get the hang of this stuff, after constantly and constantly reading over notes and trying examples. Quick question

that is how we all learned mathematics.

In one example it says,
Let U = { X in R^n | AX = 0 (Zero Vector)}
Is U a subspace?
Am I correct in reading the question as, X is in R^n if and only if AX = 0.

not quite, you should read it as X is in U if AX=0, where A is some fixed matrix, or that U is the subset of vectors x in R^n such that Ax=0. You need to show that U satisfies the axioms of a subspace, ie 0 is in U, and if x and y are in U that x+y is in U, and that if x is in U and t is a real number that tx is in U.

That is, when checking all three conditions, I should get the zero vector?

no, that doesn't follow at all

 

[JasonRox]

Thnx! I think I am starting to get the hang of this stuff, after constantly and constantly reading over notes and trying examples. Quick question. In one example it says,
Let U = { X in R^n | AX = 0 (Zero Vector)}
Is U a subspace?
Am I correct in reading the question as, X is in R^n if and only if AX = 0. That is, when checking all three conditions, I should get the zero vector?

Are you talking about the all the homogeneous solutions for the matrix A?

If that is the case, why not try yourself and see if it is a subspace. All you need to do is check if it is closed under addition and scalar multiplication.

In order words, show that if x,y is in U, then x+y is in U, and if k is any scalar, then kx is in U.

Note: If it has only one solution for the homegeneous system, then we know it is the trivial solution. The trivial solution is the zero vector itself, which creates a subspace all on it's own.