Is $S=\{(1,t)\mid t\in \mathbb{R}\}$ a subspace of $\mathbb{R}^2$? My professor introduced subspaces of $\mathbb{R}^n$ today and I don't think I understand them very well.
He posed this question as an example:

Is the set $S=\{(1,t)\mid t\in \mathbb{R}\}$ a subspace of $\mathbb{R}^2$?

He said that it wasn't. Could anyone elaborate as to why it isn't?
Can I simply say that the zero vector, $\vec{0}=(0,0)$ can never equal $(1,t)$ and be done with it?

To show that some set is a subspace, you must show two things:


*

*You can pick any two vectors in the set, add them together, and the result is in the set.

*You can pick any vector in the set, multiply it by any scalar, and the result is in the set.


The easiest check is for the $\vec{0}$ to be in the set.  Why?  If it isn't in the set, then multiplying by zero (a scalar) results in a vector not in the set!  (Violation of condition 2.)
So, in short, you're perfectly right. ;)
