2.1 The Span Collapses

Take two vectors in space, $\mathbf{v}_1$ and $\mathbf{v}_2$ . From them you can build new vectors by scaling and adding: $u\mathbf{v}_1 + v\mathbf{v}_2$ for any numbers $u$ and $v$ . The collection of everything you can build this way is called the span of $\mathbf{v}_1$ and $\mathbf{v}_2$ . Span is the reachable set: if you are allowed to walk any distance along the $\mathbf{v}_1$ direction and any distance along the $\mathbf{v}_2$ 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 $\mathbf{v}_1 = (1, 0.5, 0.2)$ and a second vector $\mathbf{v}_2$ that moves: the slider $a$ carries $\mathbf{v}_2$ from $(0, 1, 0.4)$ at $a = 0$ steadily toward $\mathbf{v}_1$ itself at $a = 1$ .

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 $\mathbf{v}_1 = (1, 0.5, 0.2)$ and the moving vector $\mathbf{v}_2(a)$ . The red line is the span of $\mathbf{v}_1$ alone. As $a$ approaches $1$ , the second direction dies and the plane collapses onto the line.

Explore the figure

At $a = 0$ , orbit the plane. Confirm it passes through the origin — the combination with $u = 0$ , $v = 0$ is always reachable.
Find the red line in the plane. Every point of it is a multiple of $\mathbf{v}_1$ : 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 $a = 1$ . How much of space is reachable now? What was lost, and what remained?

What collapsed, exactly

At $a = 1$ the second vector equals the first, so the combination $u\mathbf{v}_1 + v\mathbf{v}_2$ is just $(u + v)\mathbf{v}_1$ — 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.

1

The collapse slider value, solved ✓ Computed · mojocas 0.1.0 ✓ Agrees with the text

The parallelism condition from the worked example, handed to a computer algebra system: one equation, one root, one collapse. Dependence really is a single-slider-value event.

2

Two independent directions at a = 0 ✓ Computed · mojocas 0.1.0 ✓ Agrees with the text

The section's title claims a collapse, and a collapse is a change in something. That something is rank: the number of independent directions the pair actually contains, as opposed to the number of vectors listed. Park the slider at $a = 0$ and the answer is $2$ — the sheet in the figure is honestly two-dimensional.

1

One direction at a = 1 ✓ Computed · mojocas 0.1.0 ✓ Agrees with the text

The same pair at $a = 1$ , where $\mathbf{v}_2$ has arrived at $\mathbf{v}_1$ . Still two vectors; now one direction. Two computations, $2$ and $1$ , are the collapse the whole section describes — and they are what the phrase "the size of a span is governed not by how many vectors you list but by how many independent directions they contain" means, stated as a number a machine can check.

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.