Subspaces homework help

  • Thread starter jeffreylze
  • Start date
  • #1
44
0

Homework Statement



Show that the following are subspaces of R^m :

(a) The set of all linear combinations of the vectors (1,0,1,0) and (0,1,0,1) (of R^4)
(b) The set of all vectors of the form (a,b,a-b,a+b) of R^4


Homework Equations





The Attempt at a Solution



(a) If (1,0,1,0) and (0,1,0,1) span R^4 , they are subspaces.

(b) {a(1,0,1,1) + b(0,1,-1,1) | a,b [tex]\in[/tex]R}
= span {(1,0,1,1) , (0,1,-1,1)}

Are my answers incomplete ?
 

Answers and Replies

  • #2
283
3


To show its a subspace you need to show it is closed under the operation. What sort of space are we talking about here? Vector spaces?
 
  • #3
35,049
6,784


Homework Statement



Show that the following are subspaces of R^m :

(a) The set of all linear combinations of the vectors (1,0,1,0) and (0,1,0,1) (of R^4)
(b) The set of all vectors of the form (a,b,a-b,a+b) of R^4


Homework Equations





The Attempt at a Solution



(a) If (1,0,1,0) and (0,1,0,1) span R^4 , they are subspaces.
Two vectors can't possibly span R^4, a vector space of dimension 4. Also, the vectors themselves aren't subspaces. You are supposed to show that the set of all linear combinations of these two vectors is a subspace of R^4. To do that show that:
  1. The zero vector in R^4 is in this set (i.e., the set of linear combinations of the two vectors).
  2. Any two vectors in this set is also in the set.
  3. Any scalar multiple of a vector in this set is also in this set.
(b) {a(1,0,1,1) + b(0,1,-1,1) | a,b [tex]\in[/tex]R}
= span {(1,0,1,1) , (0,1,-1,1)}
See above.
Are my answers incomplete ?
 

Related Threads on Subspaces homework help

  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
2
Views
703
  • Last Post
Replies
5
Views
644
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
4
Views
990
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
4
Views
1K
Replies
6
Views
3K
  • Last Post
Replies
3
Views
2K
Top