(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine whether the following sets form subspaces of ℝ^{2}:

(a) {(x_{1}, x_{2})^{T}| x_{1}+ x_{2}= 0}

(b) {(x_{1}, x_{2})^{T}| x_{1}* x_{2}= 0}

2. Relevant equations

3. The attempt at a solution

I know that a is a subspace and b is not, but I would like to know why.

For part A, I let x=[c, -c]^{T}

∂[c,-c]= [∂c, -∂c]

[c, -c] + [ ∂, -∂] = [c+∂, -c-∂]

Thus S is closed under scalar multiplication and addition.

But what if I let x=[1, -1]? Wouldn't that break the conditions since

∂[1,-1]=[∂,-∂] and [1,-1] + [1, -1]= [2,-2]?

And for part B the book states "No, this is not a subspace. Every element of S has at

least one component equal to 0. The set is closed under scalar multiplication, but

not under addition. For example, both (1, 0)^{T}and (0,1)^{T}are elements of S, but their sum is not."

But can't I let [x_{1}and x_{2}] be the zero vectors and S would be a subspace?

I am confused about how sometimes I can multiply or add using variables and other times I have to use constants. Can someone please explain to me. Thanks

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# Homework Help: Confused about subspaces!

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