2.1 The Span Collapses

Take two vectors in space, $𝐯 sub 1$ and $𝐯 sub 2$ . From them you can build new vectors by scaling and adding: $u 𝐯 sub 1 plus v 𝐯 sub 2$ for any numbers $u$ and $v$ . The collection of everything you can build this way is called the span of $𝐯 sub 1$ and $𝐯 sub 2$ . Span is the reachable set: if you are allowed to walk any distance along the $𝐯 sub 1$ direction and any distance along the $𝐯 sub 2$ direction, span is every place you can end up.

For most pairs of vectors the span of two vectors is a plane through the origin — two independent directions of travel sweep out a flat two-dimensional sheet. But the word "independent" is carrying weight. The figure below spans a surface by $𝐯 sub 1 equals open parenthesis 1, 0.5, 0.2 close parenthesis$ and a second vector $𝐯 sub 2$ that moves: the slider $a$ carries $𝐯 sub 2$ from $open parenthesis 0, 1, 0.4 close parenthesis$ at $a equals 0$ steadily toward $𝐯 sub 1$ itself at $a equals 1$ .

A semi-transparent parallelogram-shaped surface through the origin, spanned by two vectors, with a solid red line marking the direction of the first vector. A slider named a slowly rotates the second spanning vector toward the first; as a approaches 1 the surface tilts, narrows, and finally flattens onto the red line, showing the span collapsing from a plane to a line. — The span of $𝐯 sub 1 equals open parenthesis 1, 0.5, 0.2 close parenthesis$ and the moving vector $𝐯 sub 2 open parenthesis a close parenthesis$ . The red line is the span of $𝐯 sub 1$ alone. As $a$ approaches $1$ , the second direction dies and the plane collapses onto the line.

Explore the figure

At $a equals 0$ , orbit the plane. Confirm it passes through the origin — the combination with $u equals 0$ , $v equals 0$ is always reachable.
Find the red line in the plane. Every point of it is a multiple of $𝐯 sub 1$ : the span of one vector, sitting inside the span of two.
Drag $a$ slowly toward $1$ . Describe what happens to the sheet. At what slider value do you first notice it thinning?
Park at exactly $a equals 1$ . How much of space is reachable now? What was lost, and what remained?

What collapsed, exactly

At $a equals 1$ the second vector equals the first, so the combination $u 𝐯 sub 1 plus v 𝐯 sub 2$ is just $open parenthesis u plus v close parenthesis 𝐯 sub 1$ — a multiple of a single direction. Two vectors, but only one direction of travel: the span is the red line. We say the pair has become linearly dependent: one of the vectors contributes nothing that the other did not already provide. For every $a$ strictly less than $1$ the two vectors point in genuinely different directions, and the span is a full plane; the size of a span is governed not by how many vectors you list but by how many independent directions they contain.

Notice also what never changed during the collapse. At every value of $a$ the span contained the origin, and at every value it was closed under adding and scaling: combine two reachable points and you get a reachable point. Spans are flat things through the origin — this observation becomes the definition of a subspace two sections from now.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Its chapter sequence is aligned to Interactive Linear Algebra (Margalit & Rabinoff, Georgia Tech, GNU FDL); this work is original, copies nothing from it, and is not affiliated with or endorsed by its authors. License: CC-BY-NC-SA-4.0.