Checking Vector Space Questions

• Sep 8th 2010, 10:13 PM
acevipa
Checking Vector Space Questions
Okay, so I'm having some trouble with this topic. If a question asks you to show that a set is a subspace (or not a subspace), do you show the following:

$\displaystyle \bullet$ The set $\displaystyle S$ is not empty
$\displaystyle \bullet$ The set is closed under vector addition, i.e. $\displaystyle \mathbf{v,u}\in S \Longrightarrow \mathbf{v}+\mathbf{u}\in S$
$\displaystyle \bullet$ The set is closed under scalar multiplication, i.e. $\displaystyle \mathbf{u}\in S \ \mbox{and} \ \lambda \in \mathbb{F}\Longrightarrow \lambda\mathbf{u}\in S$

1) Show that the set

$\displaystyle S=\{\mathbf{x}\in\mathbb{R}^3:2x_1+3x_2-4x_3=6\}$

is not a subspace of $\displaystyle \mathbb{R}^3.$

Would I do the following:

$\displaystyle 2(\mathbf{0})+3(\mathbf{0})-4(\mathbf{0})=6$

Therefore, it does not contain the $\displaystyle \mathbf{0}$ vector, hence not a subspace in $\displaystyle \mathbb{R}^3$
• Sep 9th 2010, 02:59 AM
Ackbeet
Right, so if the zero vector is not included, you couldn't possible have closure under either scalar multiplication (think multiplication by zero) or vector addition (a vector plus its additive inverse should give you the zero vector). Generally yes, for a subspace, you get to inherit everything except the three properties you listed. So those are the ones you have to check.