I\'ve really been struggling with this subspace and theres a few problems I want

Question

I've really been struggling with this subspace and theres a few problems I want to see if I've answered them correctly.

1. The set S = {(x1, x2, x3} where X1=X2, and X2=2X3} is a subspace of R4.

I said no because S in R3 not R4.


2. S = {(x1, x2} where x1x2 >or= 0} is a subspace of R2.

I said no because it is not closed under the property of addition. But I'm uncertain on this...

3. S = {(x1, x2, x3) where x1+2(x2)+x3=4} is a subspace of R3.

I said yes to this one.

4.) S = {(-a+2b, a+b, a+3b) where a and b are any numbers} in R3.

Yes to this one as well.

Any input is greatly appreciated! Thanks!
I've really been struggling with this subspace and theres a few problems I want to see if I've answered them correctly.

1. The set S = {(x1, x2, x3} where X1=X2, and X2=2X3} is a subspace of R4.

I said no because S in R3 not R4.


2. S = {(x1, x2} where x1x2 >or= 0} is a subspace of R2.

I said no because it is not closed under the property of addition. But I'm uncertain on this...

3. S = {(x1, x2, x3) where x1+2(x2)+x3=4} is a subspace of R3.

I said yes to this one.

4.) S = {(-a+2b, a+b, a+3b) where a and b are any numbers} in R3.

Yes to this one as well.

Any input is greatly appreciated! Thanks!

Explanation / Answer

(1). R3 is a subspace of R4. Therefore the answer is Yes.

(2). This answer is correct due to the closure property of addition.

(3) The answer is no. As points (1,1,1) and (1,2,-1) are in S but there sum is not. Therefore due to closure of addition.

(4). The answer is yes.

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