# 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?

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Is the set of all linear combinations of $(1,0,0)$ and $(0,1,1)$ a vector space? If so, what is its dimension? Once you figure out its dimension, there are two ways to present a vector basis -- as a linear combination of vectors or as the solution of a set of (homogeneous) linear equations. Can you find the linear equations that describe this set of linear combinations? – Michael Joyce Feb 17 '12 at 18:30

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.
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.