I just started taking linear algebra and I am already confused about how to know whether linear combinations fill a line, plane, or $R^3$, my textbook simply says it you have one vector with a scalar multiple it fills a line, if you add two vectors with scalar multiples it fills a plane, and a 3 vector linear combination fills $R^3$, it also says to be careful when zeros are involved because it's not so obvious but then doesn't explain how to figure it out. Here are the two questions I am having touble with.

A. $u=(1, 0, 0) \text { and } v=(0, 2, 3)$
would this fill a line because you are never adding both x, y, or z values?

B. $u=(2, 0, 0), v=(0, 2, 2) \text{ and } w=(2, 2, 3)$
would this linear combination only fill a plane because when adding u and v you are never adding two of the same coordinates?


I don't know what zeroes have got to do with it.

Two vectors will always fill a plane unless one is a multiple of the other.

Three vectors will usually fill $R^3$, but you need to watch out for one of the vectors being a linear combination of the others. Given vectors $u$, $v$ and $w$, you need to check that there are no scalars $a$ and $b$ such that $w=au+bv$. If that is the case, then you only have two independent vectors and they fill a plane. Of course, if all three are multiples of each other then you just have a line.

Your questions:

A. It will fill a plane because $v \ne ku$.

B. They're not all multiples of each other, so not a line.

Try finding $au+bv=w$

Gives equations $2a=2$, $2b=2$, $2b=3$.

These are not consistent, so $w$ is not a linear combination of $u$ and $v$, so they fill $R^3$

  • $\begingroup$ thank you, the textbook does not even mention information on what happens if vectors are multiples of each other. $\endgroup$
    – idknuttin
    Feb 5 '16 at 23:03

For part A, it’s just the opposite: each vector contributes something that the other doesn’t, so together they fill a plane.

In part B you’ve got the same situation with the vectors $u$ and $v$—each makes an independent contribution to the whole. Now, does $w$ also make its own contribution, i.e., is it linearly independent of the other two vectors? Any linear combination of $u$ and $v$ must have the same value for the last two coordinates, but that’s not the case for $w$, so mixing in a multiple of $w$ will vary the last coordinate and the three vectors fill the entire space.

More generrally, it’s not so much how many of the coordinates are affected by the given vectors as it is how many will vary independently of each other. For example, multiples of a single vector will in general affect all of the coordinates, but they will vary in lock-step with each other, so you’ll only get a line that way.


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