Choose x = (x1, x2, x3, x4) in r4. it has 24 rearrangements like (x2, x1, x3, x4) and (x4, x3, x1, x2). those 24 vectors, including x itself, span a subspace s. find specific vectors x so that the dimension of s is: (a) 0, (b) 1, (c) 3, (d) 4.

a. Take to be the zero vector.

b. Fix one component from , then set for . One such choice would be .

c. Similar strategy as in (b), but we need 3 vectors in order for to have dimension of 3. One such choice would be .

d. You can use the example from (c); then you just need to insert the vector .

i think this could be a possible answer (1 / 84) -1

good luck! : d