Proving Subspaces of R^m: Linear Combinations and Vector Forms

[jeffreylze]

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 $$\in$$R}
= span {(1,0,1,1) , (0,1,-1,1)}

Are my answers incomplete ?

 

[Focus]

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?

 

[Mark44]

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 $$\in$$R}
= span {(1,0,1,1) , (0,1,-1,1)}

See above.

Are my answers incomplete ?