Sets and vector subspaces

Question

Progressing through my linear algebra learning, I stumbled accross vector spaces. I know that for a set in V to be a vector subspace of V, it must satisfy the 3 following conditions:

i) 0 e U.

ii) if u1, u2 e U, then u1+u2 e U.

iii) if u e U, then $\alpha$ u e U.

Are the following sets of R^4 subspaces of V?

1. U1 = {(0, 0, 0, 0)}

Here, I can see that first condition is satisfied and that this set is the smallest vector subspace of V. I also know that any linear combinations of scalars will remain in V.

2. U2 = {(x1, x2, x3, x4) | x2x3 = x1}

If x2 and x3 belong to V, then their combination will be. This is something I'm not sure to understand, but does it mean that x1 is a combination of x2x3, thus it is also in V?

3. U3 = {(x1, x2, x3, x4) | x1 = x2 + x3, x2 = 3 x4}

x1 = x2 + x3 should work because it satisfies the 2nd condition and x2 = 3 x4 also because 3 is a scalar that belongs to R^4.

Is this right? What should I add to demonstrate this properly?

Answer 1

You need to make sure that you are not confusing the closure properties ( ii) and iii) ) with the definition of the subset.

Question 2. asks whether $U_2 = \{(x_1, x_2, x_3, x_4)\in \mathbb{R}^4 |\ x_2x_3 = x_1 \}$ is a subspace, so you are asked are i), ii), and iii) satisfied?

i) $0\in U_2$ because testing whether $0\in U_2$ is testing whether $x_2x_3 = x_1$ which for $0=(0,0,0,0)$ is testing whether $0\cdot 0 = 0$.

ii) asks for $x,y\in U_2$ is $x+y\in U_2$. That is, given $x_2x_3 = x_1$ and $y_2y_3 = y_1$ is $(x_2+y_2)(x_3+y_3) = x_1+y_1$?

iii) is tested similarly using the definitions for $U_2$ and the operations inherited from the original vector space.

You should be able to come up with a counter-example to ii) or iii) for $U_2$.

Question 3. follows the same thinking. The conditions defining $U_3$ are $x_1 = x_2 + x_3$ and $x_2 = 3x_4$. You need to decide

i) Does $(0,0,0,0)$ satisfy those conditions?

ii) If $(x_1, x_2, x_3, x_4)$ and $(y_1, y_2, y_3, y_4)$ satisfy the conditions, does $(x_1+y_1, x_2+y_2, x_3+y_3, x_4+y_4)$?

iii) If $(x_1, x_2, x_3, x_4)$ satisfies the conditions, does $(\alpha x_1, \alpha x_2, \alpha x_3, \alpha x_4)$ for every $\alpha \in\mathbb{R}$?

Answer 2

Every subspace must always have the vector $ 0 $ in it. Does the set U3 have the vector $ 0 $ in it?