Geometrically describing linear combinations I am looking to describe geometrically (as a line, plane,...) all linear combinations of the following vectors-
$(1, 0, 0)$
$(0, 1, 1)$
so if to get all linear combinations I take c(1,0,0) + d(0,1,1) = (c, d, d)
This looks to me like it 'hits' every point in $R^3$ but only in the form of (c,d,d). So a point (5,9,7) isnt the set of combinations. So what is the geometric description for the set of all linear combinations?
 A: To add on to the other answers already given, here is a diagram showing the plane that the vectors span. For some intuition, the linear combinations which have integer coefficients (eg, 3(1,0,0)+2(0,1,1)) will be exactly the lattice points where the red and green lines meet, while the set of all possible (fractional, irrational, etc) linear combinations will be the whole plane shown.

A: You have to evaluate the $w=u\times v$ then what you obtain by doing these linear combination is the plane with $w$ as a normal vector.
A: Picture the line $y=z$ in $\mathbb{R}^3$.  This will be a subset of your set, since all points are of the form $(c,d,d)$.  Then, since $x$ varies independently of $y$ and $z$, the $x$-coordinate of every point can take on any value in $\mathbb{R}$.  So, imagine taking the line $y=z$ and stretching it infinitely in the positive and negative directions along the $x$-axis.  The resulting plane is your set.
A: There is another geometric meaning to the term "linear combination". Given two points in space p1 and p2, the linear combinations are all those points generated by a scalar parameter t are:
p <--  t * p1  + (1-t) * p2   (-inf < t < +inf)
The set of points p then fall on the straight line determined by p1 and p2.  The term "linear" in this case indicates combinations forming a line.
This is different from a linear combination (spanning set) of vectors, where there is full independence in the choice of coefficients applied to the vectors. This seems to be the problem you're asking about.
If you were to just start generating some points, e.g., 
[ 2 , -3 , -3 ] 
[ -4 ,  1 , 1  ]
...
and plotting them in 3D, I think you'll be able to solve this.
