Which of the following subsets of R3 are actually subspaces

Which of the following subsets of R3 are actually subspaces? (a) The plane of vectors (b1,b2,b3) with rst component b1 =0. (b) The plane of vectors b with b1 =1. (c) The vectors b with b2 b3 =0 (this is the union of two subspaces, the plane b2 =0 and the plane b3 =0). (d) All combinations of two given vectors (1,1,0) and (2,0,1). (e) The plane of vectors (b1,b2,b3) that satisfy b3b2+3b1 =0.

Solution

This set is of type i) and is therefore a subspace of R^3.

5) The phrase \"linear vectors\" is confusing, so I will treat this phrase as \"vectors\" or \"plane of vectors\" (note that b1+b2+b3=0 is a plane). This set is of type ii) and is therefore a subspace of R^3.

6) The inequality should make us suspect that this set is not a subspace. This set violates condition c) because (1,2,3) is in the set (since 1 <= 2 <= 3), but its negative, (-1,-2,-3), is not in the set since (-1 > -2 > -3). Therefore, this set indeed is not a subspace of R^3.