Span and Independence: When the Dimension Jumps

Start with the red arrow v zero. Its multiples, c zero times v zero, fill one line through the origin: the span of one vector.

Add the blue arrow v one, off that line. Now the combinations fill the purple plane: dimension two.

The green arrow v two sticks out of the plane. The three arrows build a box with a real volume, and their combinations reach all of space: dimension three.

Watch v two sink toward the plane. The box flattens, and its volume shrinks smoothly. The dimension stays three until v two touches the plane, then drops to two, all at once.

Inside the plane, v two is a combination of the other two: one half of v zero plus one half of v one. So the three vectors are linearly dependent.

Your turn. Pull v two out of the plane, then put it back. Drag its green tip up or down, or its shadow on the floor. Near the plane, it snaps in.

The volume changes smoothly, but the dimension only jumps: vectors are independent or they are not. Move v two anywhere, and watch both numbers.