Just like I enjoy using Desmos activities when appropriate, I believe in giving students an opportunity to explore visual concepts in an interactive setting. This is especially useful in geometric classes such as Linear Algebra.

Linear Algebra: Visualizing Span

This Geogebra interactive is designed to supplement chapter 1.3 of Lay: Vector Equations. The text introduces the concept of span, including some geometric intuition, but students expressed confusion about the geometric representation of the span of two vectors being (in most cases) a plane.

I suggest the following activities to supplement this interactive:

In this visualization, I show you the vectors \(\vec{v}_1\) and \(\vec{v}_2\) (in red and blue) together with a linear combination \(\vec{w} = c_1 \vec{v}_1 + c_2 \vec{v}_2\) (in purple, get it? because it’s a combination of red and blue). Pressing “play” on the scalar multiples \(c_1\) and \(c_2\), you can look at many different possible linear combinations of \(\vec{v}_1\) and \(\vec{v}_2\), each given by the purple vector (parallelogram rule for addition visualized as well).

  1. How do the linear combinations relate to the purple plane, which represents the “span” of the two vectors?
  2. Try changing \(\vec{v}_1\) and \(\vec{v}_2\) to other vectors of your choosing. How does the span change when you change the two vectors?
  3. What happens if you pick \(\vec{v}_1 = (0,1,1)\) and then set \(\vec{v}_2 = (0,2,2)\)? What does the “span” look like, then? Can you think of a general rule for when this would occur?