Exercise 3.4.16

These bases are not unique!

(a)

$(1,1,1,1)$ for the space of all constant vectors $(c,c,c,c)$

(b)

$(1,-1,0,0),(1,0,-1,0),(1,0,0,-1)$ for the space of vectors with sum of components $=0$

(c)

$(1,-1,-1,0),(1,-1,0,-1)$ for the space perpendicular to $(1,1,0,0)$ and $(1,0,1,1)$

(d)

The columns of $I$ are a basis for its column space, the empty set is a basis (by convention) for $N(I)=Z=\{$ zero vector $\text{}}$.