How do I know the span of a matrix I have a marix that is 
$$R=\pmatrix{1&1&1\\1&1&1+2a\\0&2&2+2b}$$
and the span of R is   span=$\left\{\pmatrix{1\\1\\0},\pmatrix{0\\0\\1}\right\}$
In my preliminery thinking, the span is the combination of vectors that indicate each vector in R, but in reverse, If I know a matrix, how can I derive the span?
Could anyone tell me I can I calculate the span of a matrix
Taking this as an example, how can I get to this result?

Your question is ambiguous, cause in general, for fixed $n,m$, the set $S=M_{n\times m}(\mathbb{K})$ (matrices of $n\times m$ with entries in the field $\mathbb{K}$) is a vector space over $\mathbb{K}$. Then, if $A\in S$, definition of $span(A)$ is the usual definition for span of a vector in $S$.
However, I suppose indeed in your problem you are asking for the column space spanned by a matrix. i.e. the space spanned for the columns of the matrix.
Then, you are searching for $V=span\{(1,1,0)^T,(1,1,2)^T,(1,1+2a,1+2b)^T\}$ and, by hypothesis, this is $span\{(1,1,0)^T,(0,0,1)^T\}$.
As vectors $(1,1,0)^T,(0,0,1)^T$ are linearly independents, you need that $(0,0,1)^T$ let a linear combination of $(1,1,2)^T,(1,1+2a,1+2b)^T$. That is, find $x,y$ such that 
$$
\left\{\begin{array}{ccc}x+y&=&0\\x+y(1+2a)&=&0\\2x+y(1+2b)&=&1\end{array}\right.
$$ 
for given $a,b$.
Solving for $x$ and $y$, we get, for $a\neq 0$, there are not $x,y$. For $a=0$, and $b=-3/2$, there are not solutions, and for $a=0$ and $b\neq -3/2$, $x=-1/(3+2b)$ and $y=1/(3+2b)$   
