This flashandmath.com applet (linked below) is another one based on “things I wave my hands at but cannot draw on the board.” I often describe the span of a set of vectors as “the points reachable by following combinations of scaled vectors in the set.” This description requires a visualization of something active: “I walk 2 times the length of u (in its direction) and then 1/2 times the length of v (in its direction), which brings me to the point at the end of the vector 2u + 1/2v.”

The appet allows the user to draw any two vectors u and v and enter a limited linear combination of the form w=au + bv. Instead of just drawing the resulting vector w, the applet shows an animation in which we follow the path from the origin, to au, and then to au + bv. Comments welcome, of course!

Open the applet by clicking here (http://webspace.ship.edu/deensley/flash/ela/linearcombinations.html) on the screenshot below:

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