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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
up vote 3 down vote accepted

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. enter image description here

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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.

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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.

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