Vector Sets and Span

Question

For each set of vectors, determine whether their span is {0}, a line, a plane, all of R², or all of R³.

A) {(0, 0), (3, 7)}

B) {(1, 2),(3, 4)}

C) {(0, 1, −2),(2, 1, 0),(1, 1, −1)}

Answer 1

The span of a set of vectors is just the points you can get to by adding different multiples of them.

A) Adding multiples of (0,0) won't get us anywhere; it doesn't contribute anything to the span except {0}. But adding multiples of (3, 7) can get us anywhere along the line defined by the origin and (3,7). So the span is a line. For example, getting to (1.5, 3.5), which is on that line, is possible with 0.5 multiples of (3, 7).

B) This time we have two different vectors in two dimensions. If the two vectors are perfectly parallel, then we would still have a line, for example if it were (1.5, 3.5) and (3, 7) we would still have the same line as before because (1.5, 3.5) was already reachable. When one of your vectors can be reached using multiples of the others, they are called linearly dependent. But in this case, there is no way to make (3,4) by multiplying (1, 2) by a constant and your vectors are not linearly dependent. When your vectors are not linearly dependent, you can do the following interesting thing:

(1, 0) = a * (1,2) + b * (3, 4) implies 3b + a = 1 and 2a + 4b = 0 so a = -2b and b = 1 and = -2

If we take 1 copy of the second vector and two of the first, we can make (1, 0)

(0,1) = c * (1,2) + d * (3,4) implies c + 3d = 0 and 2c + 4d = 1 so c = -3d and d = -.5 and c = 1.5

If we take 1.5 copies of the first vector and -.5 copies of the second, we can make (0,1)

Since can make (1, 0) and (0, 1), we can make any point that (1,0) and (0, 1) can make.

And (1,0) and (0,1) can make the whole x-y plane. Since we are only in two dimensions to begin with, we call this plane all of R²

This turns out to work for any two vectors that are not linearly dependent.

C) Sorry, in a bit of a hurry.

Adding the first two vectors gives twice the third vector. That means the vectors are linearly dependent. But the first two are not dependent on each other, so you can still make a Plane But it would take a third independent vector, which you don't have, to make a 3 dimensional space (all of R³).

In general, with 1 independent vector you get a line. With two you get a plane. With 3 you get a 3d space like the one we live in. And you can keep going but it becomes harder to visualize. Also, if your number of independent vectors equals your number of variables in each vector, you will span your whole space, which can be R² or R³ or something else depending on the number of variables. With N independent vectors with values for N different variables you span R^N

^{Hope that helps!}