Which of the following are true So I'm taking math in French, and my teacher refers to something that would directly translate to : Linear envelope (I've been told it's linear span in english!), but when I google that, I get nothing. I really don't understand it, and if someone could point me to the right ressources, that would be great. I'm sure it's easy to deduce what he's referencing as I have the solution, but I just have no idea why its correct.
Which of the following two statements are true!
1- The linear span of two distincts vecters U and V in $R^3$ is a plane that passes by the origin. (answer is false, he says it's a line)
2- the linear span of one vector u in $R^2$ is a line. (answer is false, because if  u = 0, the span of {u}={o})
3- the set of vectoers {u,v,w} begets a vector space X if every vector x element of X is a linear combination of v and w. (answer is true X = span{v,w} which is a subset of span{v,w,u} = X
4- the set of {(1,0),(1,1,)} begets $R^2$ (It's true, I think he is trying to say that it's spanning all of $R^2$?) he says that (0,1) = (1,1) - (1,0) and thus (0,1) is element of the span{(1,0),(1,1)} which is element of R^2
5- the set {$\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}$ , $\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}$, $\begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix}$} begets the vector space M2(R) of the matrixes in format (2x2). (wrong because $\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}$ is not an element of the span of the set)
 A: If he really is referring to the span, then here would be a brief discussion:
Defn: The span of a set of vectors is defined by $span \{\vec{v}_1 , \vec{v}_2, \dots \vec{v}_n \} := \{ a_1 \vec{v}_1 + a_2 \vec{v}_2 + \cdots + a_n \vec{v}_n ~|~ a_i \in \mathbb{F} \} = \{ \sum a_i \vec{v}_i ~|~a_i \in \mathbb{F} \}$, where $\mathbb{F}$ is the appropriate scalar field. The span of a set of vectors is the smallest vector space that contains all the listed vectors.
In particular, I want to use this definition in the context of the posed / questions answers, interpreting the phrasing as best as I can. 


*

*$span \{\vec{u}, \vec{v}\}$, where $\vec{u}, \vec{v} \in \mathbb{R}^3$ are linearly independent, is a plane that includes the origin--hence false. Notice that $0 \vec{u} + 0 \vec{v} = \vec{0}$, regardless of what $\vec{u}$, $ \vec{v}$ are. 

*If $\vec{u} = \vec{0}$, then $span \{ \vec{0} \} = \{ a \vec{0} ~|~a \in \mathbb{R} \}=  \{\vec{0} \}$. If $\vec{u} \neq \vec{0}$, then $span \{\vec{u}\}$ generates a line through the origin. 

*The set of vectors $\{ \vec{u}, \vec{v} , \vec{w} \}$ generates a vector space $X$ if every vector $\vec{x} \in X$ can be written as a linear combination of  $\vec{u}, \vec{v} , \vec{w}$. That is, $X = span \{ \vec{u}, \vec{v} , \vec{w}\}$. Notice that there is no requirement of linear independence, just that there are enough vectors to generate everything in the vector space. E.g., $\mathbb{R}^2 = span \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \end{bmatrix} \}= span \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 5 \\ 2 \end{bmatrix}\}$

*The set $\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \end{bmatrix} \}$ does in fact generate $\mathbb{R}^2$: every vector $\vec{v} \in \mathbb{R}^2$ can be written as a linear combination of these two vectors. We could say that this is a spanning set, that $span \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \end{bmatrix} \} = \mathbb{R}^2$, or even that $\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \end{bmatrix} \}$ generates $\mathbb{R}^2$. All are reasonable phrasings; the second is probably the most precise. 

*$span \{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} 0 & 1 \\ 0 &0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} \}\neq M_{2\times 2}(\mathbb{R}) = \{\textrm{set of all 2x2 matrices}\}$ for precisely the reason noted--the matrix $\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$ cannot be written as a linear combination of our three matrices. Note, however, that taking the span of these three matrices generates a vector subspace of $M_{2 \times 2 } (\mathbb{R})$, just not the entire space. 
Hope that helps out--hearing a new term applied in context always helps me understand what the term is capturing. 
