# Thread: Showing A Subset Is A Subspace

1. ## Showing A Subset Is A Subspace

Which of the following subsets of C^3 (complex) are subspaces:

a) {(a,b,c) in C^3 | c=0}

b) {(a,b,c) in C^3 | |a| is less than or equal to |b|}

c) {(a,b,c) in C^3 | |a^2| + |b^2| + |c^2| =0}

I'm having trouble understanding how to use vector addiction and scalar multiplication to see if they are subspaces or not. Any help is appreciated.

2. Are you familiar with this criteria for vector subspaces:

Let $\displaystyle W\subseteq V$ be a subset of a vector space $\displaystyle V$ over a field $\displaystyle F$. $\displaystyle W$ is subspace of $\displaystyle V$ if and only if:
1. $\displaystyle W\neq \emptyset$.
2. For all $\displaystyle w_{1},w_{2}\in W$, $\displaystyle \lambda _{1},\lambda _{2}\in F$ exists $\displaystyle \lambda _{1}w_{1} + \lambda _{2}w_{2}\in W$.

Try to use this for your problem.
Here's an example for the first part:

First you determine that the given set is not empty, well clearly $\displaystyle \theta$ is of the form $\displaystyle (a,b,0)$, so $\displaystyle \theta \in A$ so $\displaystyle A \neq \emptyset$.

Second, let $\displaystyle w_{1}=(a,b,0), w_{2}=(d,e,0)\in A$ be some arbitrary vectors and let $\displaystyle \lambda _{1},\labmda _{2}$ be some complex scalars. now $\displaystyle \lambda _{1}w_{1} + \lambda _{2}w_{2} = (\lambda _{1}a+\lambda _{2}d,\lambda _{1}b+\lambda _{2}e,0)\in W$.