Jump to content
  • Advertisement
Sign in to follow this  
NerdInHisShoe

Linear algebra - subspaces

This topic is 4227 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

If you intended to correct an error in the post then please contact us.

Recommended Posts

I'm a bit confused about Linear spaces, particularly subspaces. One of the conditions of a vector space is that there is a zero vector such that x + 0 = x. For example, would the plane x + 2y - 3z = 1 fail to be a subspace of R^3 because (0,0,0) is not on the plane, or am I just way off? Thanks :)

Share this post


Link to post
Share on other sites
Advertisement
Quote:
For example, would the plane x + 2y - 3z = 1 fail to be a subspace of R^3 because (0,0,0) is not on the plane, or am I just way off?

You are correct. For a plane to be a subspace of R^3 it has to be of the form Ax+By+Cz=0 .

Share this post


Link to post
Share on other sites
The zero vector just says that p + 0 = p, which holds for both R^3 and R^2 so
(0,0,0) isn't necessary for the subspace to be R^2.

Share this post


Link to post
Share on other sites
Quote:
Original post by Alrecenk
Quote:
For example, would the plane x + 2y - 3z = 1 fail to be a subspace of R^3 because (0,0,0) is not on the plane, or am I just way off?

You are correct. For a plane to be a subspace of R^3 it has to be of the form Ax+By+Cz=0 .


And likewise, a subset of the set of all polynomials would not be a subspace unless the zero polynomial was an element?

Share this post


Link to post
Share on other sites
Quote:
Original post by NerdInHisShoe
And likewise, a subset of the set of all polynomials would not be a subspace unless the zero polynomial was an element?


Exactly. Keep in mind that the requirement of a zero element comes directly from the fact that a subspace is closed under scalar-vector multiplication, and the underlying field has a zero element. If x is in a subspace, then x*0 = 0 is in a subspace.

Share this post


Link to post
Share on other sites
If x and y are elements of a vector space, then x + y must also be an element, does that mean that vector spaces must have an infinite number of elements, well, except for the zero vector space I suppose?

Share this post


Link to post
Share on other sites
Quote:
Original post by NerdInHisShoe
If x and y are elements of a vector space, then x + y must also be an element, does that mean that vector spaces must have an infinite number of elements, well, except for the zero vector space I suppose?


That depends on whether the space's underlying field has infinitely many elements. In the real numbers, for example, the vector space would need to be infinite (except for the "trivial" vector space, as you said), but under some modular field (i.e. the integers modulo p), the space would not have to be infinite.

Share this post


Link to post
Share on other sites
Quote:
If x and y are elements of a vector space, then x + y must also be an element, does that mean that vector spaces must have an infinite number of elements, well, except for the zero vector space I suppose?


Not necessarily. The underlying field may have a finite number of elements; for example, all Zp (that is, integers modulo p), where p is a prime, form a field. See Modular arithmetic, Galois field.

Share this post


Link to post
Share on other sites
It depends upon how you define + and * in your space. If you define them in the regular way then your space has to be infinite, but it's possible to define them such that that is not the case.

Share this post


Link to post
Share on other sites
Sign in to follow this  

  • Advertisement
×

Important Information

By using GameDev.net, you agree to our community Guidelines, Terms of Use, and Privacy Policy.

GameDev.net is your game development community. Create an account for your GameDev Portfolio and participate in the largest developer community in the games industry.

Sign me up!