# Thread: Independent v.s. dependent columns

1. ## Independent v.s. dependent columns

Hi,

I'm lost with this image:

Can someone please explain the description of Figure 4.7? How is a1, a2, and a3 not in a plane while c1, c2, and c3 in the same plane? I struggling to find the reasoning behind this.

(Sorry I forgot to convert the "1" in "a1" and the "2" and "a2" etc as subscripts)

Appreciate help!

- Olivia

2. ## Re: Independent v.s. dependent columns

what exactly don't you understand?

The $a's$ are stated to be linearly independent and thus they span the entire 3D space

The $c's$ are stated to be linearly dependent and thus they span a 2D space

I can see that the drawing of the $c's$ looks a bit odd but just accept that they all lie in the same plane.