math111_logo Span

2. Geometric intuition of span

The span of one vector is all the scalar multiples of the vector.

span{u} = {cu: cR} = { a line, if u0
a point, if u = 0

The span of two vectors is

span{u, v} = {cu + dv: c, dR} = { a plane, if u, v are not parallel
a line, if u, v are parallel and at least one nonzero
a point, if u = v = 0

where parallel means that one vector is a scalar multiple of another. The following picture illustrates the span of two non-parallel vectors.

If v = au, then cu + dv = (c + da)u, so that span{u, v} = span{u}, which we know is a point or a line, depending on whether u is zero or not.

The intuition continues with the span of three vectors, which can be 0, 1, 2, or 3 dimensional. When no one vector is the linear combination of the other two (or independent, a concept we will study), the span is 3 dimensional. The following picture illustrates the situation.

If one vector is a linear combination of the other two, say w = au + bv, then cu + bv + ew = (c + ea)u + (d + eb)v, so that span{u, v, w} = span{u, v}, which we know can be a plane, a line, or a point.

Intuitively, span is all the places one can go to by combining the given directions. For example, we can go from the origin to 2u - 3v by first moving in u-direction by 2 units and then moving in the opposite of v-direction by 3 units (or moving in v-direction by - 3 units). The following pictures illustrate such intuition.

The intuition explains why w = au + bv implies span{u, v, w} = span{u, v}. The reason is that any use of the direction w may be substituted by the use of the other directions. Thus w is a "wasted" direction, and deleting w should not affect the span.

To draw an analogue of the situation, since by walking and swimming, we may (at least theoretically) reach anywhere on the face of the earth, we have

span{walking, swimming} = face of the earth.

Now the addition of driving a car would certainly make your adventures more efficient. However, it will not bring you to more places. We still have

span{walking, swimming, driving} = face of the earth.

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