Subspace and Space

Question

The space $S$ is called "subspace" of $\mathbb R^2$, and this seems to me senseful because space $S$ cannot enclose all vectors in $\mathbb R^2$.

$$S = \{ (x_1, x_2)^T | 2x_1 = x_2 \}$$

I wonder if space $T$ is subspace of $R^3$ or not :

$$T = \text{span}(v_1, v_2, v_3)\\ v_1 = (1,1,1)^T, v_2 = (1,1,0)^T, v_3 = (1,0,0)^T$$

If we solve linear equations, we face these three vectors really spans $\mathbb R^3$. Hence, $T$ is not subspace of $\mathbb R^3$. I think $T$ is $\mathbb R^3$ itself.

Answer 1

A subset $U\subseteq V$ of a vector space $V$ is called a subspace iff:

• $0 \in U$
• for every $v,w\in U$ and every scalar $\alpha$, we have $\alpha v+w\in U$

In particular, every vector space $V$ is a subspace of itself (it's exactly the same thing as in set theory - each set is a subset of itself).

In your case $T$ is just $\mathbb R^3$, so it is a subspace.