So I have
\begin{align*}
A&= \{(x_1,x_2)'\in \mathbb{R}^2: \max (|x_1|,|x_2|) \leq 1\} \\
B &= \{(x_1,x_2,x_3,x_4)'\in \mathbb{R}^4:x_1⋅x_2=0\} \\
C&=\{(x_1,x_2,x_3)' \in \mathbb{R}^3 :x_2-x_3-1=0 \}
\end{align*}
And i want to determine whether these subsets of $\mathbb{R}^n$ are vector subspaces or not.
I know that $V \subset \mathbb{R}^n$ is a subspace when : the $0$ vector is in $V$, $V$ is closed under addition and $V$ is closed under multiplication .
I also learned that if $V$ is the set of solutions to homogenoeus linear equations , then $V$ is a subspace, and if $V$ is the span of some vectors , then $V$ is a subspace.
Since I have dyscalculia, I really do not know how to apply this information in order to check whether they are subspaces or not… They don't make sense to me unfortunately . Moroever, i don't know which of these concepts i should apply to which subset.
If anybody could help me and show me how these sets are subspaces or not , in the clearest and most simple way possible, that would be great ! Thank you !
Best Answer
$A$ fails to be a subspace because it is not closed under scalar multiplication. For example, $(1,1) \in A$ but $2(1,1) =(2,2) \notin A$.
$B$ fails to be a subspace because it is not closed under addition. For example, $(0,1,0,0) \in B$ and $(1,0,0,0) \in B$ but $(0,1,0,0) + (1,0,0,0) = (1,1,0,0) \notin B$.
$C$ fails to be a subspace because $(0,0,0) \notin C$.