and are parallel; paralelogram they span has area 0If , what can you say about ?

There's gotta be more?? Please tell me.

Are and meant as orthonormal vectors and ?If and , what's ?

If so, is this right: ??? Geometrically sounds right, in right-handed system , but I want to be sure.

If and what is ?

Please, check my scalar products:Are in-dependent? Do they make a basis in

So, they are in-dependent.

But, they don't make a basis because "there's not enough of them" () for 4D vector space; they don't span the whole space? OK, what's more rigorous answer?

InLet and be rotation of 90° around origin. Definekernel, imageandrankof map .kernelis only , right?

Imageis the whole ???

Rankof must be 2, otherwise it couldn't rotate (how could I better formalise my answer; if the 2 is correct of course)?

What should mean (what are and , obviously not points)?!

???Let be linear sub-space inside . Define orthogonal projection of to

???(Connects with previous quote.) Let also be ortho-normal basis for . Express projection (from previous) with this basis.