What is a Subspace

we know from 1.2 Span if a span is a line it would be a line in the $R^{n}$
- and to get a subspace it has to span an area and has to be linearly independent from one another
- a subspace can be thought of as a ‘line/plane’ that passes through the origen
  - intuitively: a subspace is a span of some vectors
the formal definition
- it is a set of vectors in $R^{n}$ which is closed under linear combinations
  - (where if you take all possible linear combitinations we get the same set of numbers)
We can show that somthing isnt a subspace using Counterexamples

Examples

S=(\begin{bmatrix} 2t \\ t \end{bmatrix}|t\in\mathbb{R}) $$ - is a subpsace of $\mathbb{R}^2$ because

let\begin{bmatrix} 2t_1 \ t_1 \end{bmatrix},\begin{bmatrix} 2t_2 \ t_2 \end{bmatrix}\in \mathbb{R}

- then #### Ex2

S=([x_{1},x_{2},x_{3}]|x_{1}+x_{2}+x_{3}=0)

- is a subspace of $\mathbb{R}^3$, because -