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Determining a subspace?

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  • #1
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Hey, this isn't for homework per se, but if anyone could lend me a hand figuring this out I'd appreciate it a lot!

Homework Statement



Determine whether Q is a subspace of R2/R3 in the following cases:

Homework Equations



Q = [tex]\{\left v = \left( v1, v2, 0 \right) | v1,v2 \in R \right\}[/tex]

Q = [tex]\{\left v = \left( v1, 0, 0 \right) | v1 \in R \right\}[/tex]

Q = [tex]\{\left v = \left( v1, v2 \right) \in R2 | v1 = v2 + 1, v1,v2 \in R \right\}[/tex]

The Attempt at a Solution



I honestly am not sure where I'm meant to start here. I know there are 3 conditions where a subspace may be valid; when it's equal to zero, If X and Y are in U, then X+Y is also in U
and If X is in U then aX is in U for every real number a.

How exactly am I meant to go about applying those rules? If someone could set me off on the right track that'd be awesome.
 

Answers and Replies

  • #2
[tex]\{\left v = \left( v1, v2, 0 \right) | v1,v2 \in R \right\}[/tex]

(0,0,0) is clearly in Q

Let x = (x1,x2,0) and y =(y1,y2,0) be vectors in Q

then x+y = (x1+y1, x2+y2, 0) is also in Q

and ax = a(x1,x2, 0) = (ax1, ax2, 0) is in Q
 
  • #3
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Hey, this isn't for homework per se, but if anyone could lend me a hand figuring this out I'd appreciate it a lot!

Homework Statement



Determine whether Q is a subspace of R2/R3 in the following cases:

Homework Equations



Q = [tex]\{\left v = \left( v1, v2, 0 \right) | v1,v2 \in R \right\}[/tex]

Q = [tex]\{\left v = \left( v1, 0, 0 \right) | v1 \in R \right\}[/tex]

Q = [tex]\{\left v = \left( v1, v2 \right) \in R2 | v1 = v2 + 1, v1,v2 \in R \right\}[/tex]

The Attempt at a Solution



I honestly am not sure where I'm meant to start here. I know there are 3 conditions where a subspace may be valid; when it's equal to zero, If X and Y are in U, then X+Y is also in U
and If X is in U then aX is in U for every real number a.
To correct your terminology, there are 3 conditions for verifying that a subset U of a vector space V is a subspace of that vector space. 1) Zero is an element of U. The other two are fine.
How exactly am I meant to go about applying those rules? If someone could set me off on the right track that'd be awesome.
See RandomVariable's reply.
 
  • #4
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Cool, thanks very much guys. Makes sense now!
 

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