Redundant Vectors

Linear Algebra (Matrices) · Axiom Academy

Some vectors pull their weight; some are dead weight. The test is whether a new vector reaches anywhere you couldn't already get to with the ones you have. The whole idea, in one picture You hand a vector to a set that already has some. Did it actually add anything? The honest test isn't how the vector looks — it's whether it reaches somewhere the old vectors couldn't already reach. If it lands inside what they already cover, it's redundant . If it breaks out, it adds a brand-new direction. Watch two vectors, and , span a flat plane — the shaded floor is every point you can build from them. Then a third vector arrives. First it lands on the floor : the span doesn't budge — redundant. Then a different third vector lifts off the floor : the span inflates from a flat sheet into all of 3D — a genuinely new direction. A vector is redundant exactly when it already lives in the span — its arrival leaves the reachable set unchanged. is redundant (linearly dependent on the rest) when — it's some combination . Then : same set, same dimension. Lift it off the floor — or don't Drag the third vector's tip around the floor, and use the height control to raise it off the plane. While it lies flat ( z = 0 ) it's stuck inside the span — redundant, the reachable set stays a plane. The instant you lift it ( ) it points somewhere new, and the span swells to fill all of 3D. Flat on the floor = redundant. Lifted off it = a new direction that grows the span's dimension.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.