Is Q a Subspace of R2 or R3?

[Tweet]

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 = $$\{\left v = \left( v1, v2, 0 \right) | v1,v2 \in R \right\}$$

Q = $$\{\left v = \left( v1, 0, 0 \right) | v1 \in R \right\}$$

Q = $$\{\left v = \left( v1, v2 \right) \in R2 | v1 = v2 + 1, v1,v2 \in R \right\}$$

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.

 

[Random Variable]

$$\{\left v = \left( v1, v2, 0 \right) | v1,v2 \in R \right\}$$

(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

 

[Mark44]

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 = $$\{\left v = \left( v1, v2, 0 \right) | v1,v2 \in R \right\}$$

Q = $$\{\left v = \left( v1, 0, 0 \right) | v1 \in R \right\}$$

Q = $$\{\left v = \left( v1, v2 \right) \in R2 | v1 = v2 + 1, v1,v2 \in R \right\}$$

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.

 

[Tweet]

Cool, thanks very much guys. Makes sense now!