Answer

Solution

1. 1

   $S$ and $T$ are coordinate axes in $\mathbb{R}^{2}$, both closed under addition and scalar multiplication; they are subspaces of $\mathbb{R}^{2}$. Also $S\cap T=\{(0,0)\}$, the trivial subspace.

2. 2

   However $S\cup T$ is not a subspace: $(1,0)\in S$ and $(0,1)\in T$, but $(1,0)+(0,1)=(1,1)\notin S\cup T$.

3. 3

   Hence the vector spaces among the listed objects are $S$, $T$, and $S\cap T$.

Answer structure / marking notes

Content note