[Math] How to Determine which subsets of R^3 is a subspace of R^3.

linear algebra

I have some questions about determining which subset is a subspace of R^3. Here are the questions:

a) {(x,y,z)∈ R^3 :x = 0} 
b) {(x,y,z)∈ R^3 :x + y = 0} 
c) {(x,y,z)∈ R^3 :xz = 0} 
d) {(x,y,z)∈ R^3 :y ≥ 0}
e) {(x,y,z)∈ R^3 :x = y = z}

I am familiar with the conditions that must be met in order for a subset to be a subspace:

  1. 0 ∈ R^3
  2. u+v ∈ R^3
  3. ku ∈ R^3

When I tried solving these, I thought i was doing it correctly but I checked the answers and I got them wrong. Can someone walk me through any of these problems? I've tried watching videos but find myself confused. Any help would be great!Thanks.

Best Answer

Do it like an algorithm. I'll do the first, you'll do the rest. I want to analyze $$I = \{(x,y,z) \in \Bbb R^3 \ : \ x = 0\}$$

The first condition is ${\bf 0} \in I$. Is it? Well, ${\bf 0} = (0,0,0)$ has the first coordinate $x = 0$, so yes, ${\bf 0} \in I$.

The second condition is ${\bf v},{\bf w} \in I \implies {\bf v}+{\bf w} \in I$. Now, I take two elements, ${\bf v}$ and ${\bf w}$ in $I$. I know that their first components are zero, that is, ${\bf v} = (0, v_2, v_3)$ and ${\bf w} = (0, w_2, w_3)$. Is their sum in $I$? Test it! $${\bf v} + {\bf w} = (0 + 0, v_2+w_2,v_3+w_3) = (0 , v_2+w_2,v_3+w_3)$$ Since the first component is zero, then ${\bf v} + {\bf w} \in I$.

The third condition is $k \in \Bbb R$, ${\bf v} \in I \implies k{\bf v} \in I$. Then, I take ${\bf v} \in I$. I know that it's first component is zero, that is, ${\bf v} = (0,v_2, v_3)$. Is $k{\bf v} \in I$? Compute it, like this: $$k{\bf v} = k(0,v_2,v_3) = (k0,kv_2, kv_3) = (0, kv_2, kv_3)$$ Is its first component zero? Yes, it is, then $k{\bf v} \in I$, and hence $I \leq \Bbb R^3$. Think alike for the rest.

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