Visualizing Subspaces and Subsets (in R3)

In summary, Vector Spaces and Subspaces are concepts studied in Linear Algebra, while Subsets are a broader concept. One way to visualize these concepts is by practicing problems. In Euclidean space, Rn, vector spaces are flat and the only subspaces are planes and lines containing the origin. In contrast, an egg cannot be considered a vector space because it does not follow the rules of vector addition and scalar multiplication. However, if we consider the yolk sack as a subspace, any additional objects within the yolk could be considered subspaces of the yolk sack.
  • #1
MaxMackie
6
0
I have trouble visualizing what exactly these are. Vector Space, Subset, Sub Space...
What's the difference and how can I "see" it. I'm a very visual person.
 
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  • #2
http://en.wikipedia.org/wiki/Vector_space#Definition"

Vector Spaces and their Subspaces are studied in Linear Algebra, while Subsets are a much more general formulation (subspaces are subsets of appropriate vector spaces with certain properties). You visualize these things concepts by doing lots of problems on them.

Here are some more links:
http://home.scarlet.be/~ping1339/Pvect.htm
http://www.cs.odu.edu/~toida/nerzic/content/set/basics.html
http://library.thinkquest.org/C0126820/setsubset.html
 
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  • #3
Thanks, I've read those over a couple times before and I get the math of it. But, let's say and egg.

Are any of these statements true about the egg?
-The eggs as a whole is a Vector Space and the yolk sack is a Subspace?
-If there was something else within the yolk, we could say that the yolk sack is a Vector Space and the thing inside is a Subspace.

Sorry to use an egg as an example, it just seems to fit.
Thanks!
 
  • #4
In Euclidean space, Rn, with addition and scalar multiplication defined "componentwise" ((x, y, z)+ (u, v, w)= (x+u,y+ v, z+ w) and r(x, y, z)= (rx, ry, rz)) vector spaces are flat. The sphere, \(\displaystyle x^2+ y^2+ z^2= R^2\), for example, is not a vector space because (1, 0, 0) is in that set but 2(1,0,0)= (2,0,0) is not.

The only subspaces of R3 are planes containing (0,0,0) and lines containing (0,0,0).

(Lines and planes that do NOT contain (0,0,0) are often referred to as "linear manifolds".)
 

1. What is a subspace in R3?

A subspace in R3 is a three-dimensional subset of a larger vector space. It is created by taking a linear combination of a set of vectors. In other words, a subspace is a set of vectors that can be added together and multiplied by a scalar to produce another vector within the same subspace.

2. How is a subspace different from a subset?

A subset is a smaller set of elements taken from a larger set, while a subspace is a subset that also follows specific mathematical rules. In particular, a subspace must contain the zero vector, be closed under vector addition and scalar multiplication, and span the entire space.

3. How can I visualize a subspace in R3?

One way to visualize a subspace in R3 is to plot the vectors that span the subspace and see how they relate to the axes. Another approach is to use geometric shapes, such as planes, to represent the subspace. Additionally, you can use software tools like Mathematica or MATLAB to create 3D visualizations of subspaces in R3.

4. How can I determine if a given set of vectors form a subspace in R3?

To determine if a set of vectors form a subspace in R3, you can check if they satisfy the subspace properties mentioned earlier, such as containing the zero vector, being closed under vector addition and scalar multiplication, and spanning the entire space. Additionally, you can use the rank-nullity theorem or perform Gaussian elimination to check if the vectors are linearly independent.

5. What are some real-life applications of visualizing subspaces in R3?

Visualizing subspaces in R3 can be useful in various fields, such as physics, engineering, and computer graphics. For example, in physics, subspaces can represent different modes of motion of a physical system. In engineering, subspaces can be used to represent the range of possible values for a set of variables in a system. In computer graphics, subspaces can be used to create 3D models and animations.

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