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What is the geometric description of $B = \text{span} \space \{[1, 1], [2, 6], [3, 5]\}$ in $\mathbb{R}^2$, where $[x, y]$ is a vector.

So $B = a[1, 1] + b[2, 6] + c[3, 5], a, b , c \in \mathbb{R}$

Is there a way to simply this further so I can get the geometric description?

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    What do you mean by geometric description? Are the vectors linearly dependent or not? – 2017-01-08
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    Perhaps an easier question might be: are there any vectors in $\mathbb R^2$ that are not in $B$? – 2017-01-08

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I suppose that you know that $\mathbb{R}^2$ is a vector space of dimension $2$ and this means (by definition of dimension of a vector space) that a basis in $\mathbb{R}^2$ contains at most two linearly independent vectors. Or, in other words that $\mathbb{R}^2$ the span of a couple of linearly independent vectors.

Now note that $\vec v=[1,1]$ and $\vec u= [2,6]$ are linearly independent, and, as a consequnce the span of these two vectors is $\mathbb{R}^2$ and, in particular, $\vec w[3,5]=2[1,1]+\frac{1}{2}[2,6]$ is a linear combination of $\vec v$ and $\vec u$.

So the ''geometric description'' of $B$ is the whole $\mathbb{R}^2$

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    For linear independence you need a set of vectors right? – 2017-01-08
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    You can see the definition of linear independence of a set of vectors here: https://en.wikipedia.org/wiki/Linear_independence – 2017-01-08
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Can I offer a suggestion? If you have a pair of perpendicular vectors, e.g.,

[ 3 , 5 ] and [ 5 , -3 ] then don't these offer an alternative x-y axis set (basis set)? What is the span of any such two vectors?

Looking at the given 3 vectors, can you combine any two of them (by vector add or subtract) into a resultant vector perpendicular to the other vector?

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    For 2 vectors to span $R^2$ they need not be perpendicular. – 2017-01-08