The union of two subspaces of a vector space is not necessarily a subspace.

There are four vectors (1,a,b,c), (1,1,1,1), (1,2,3,1), (3,4,5,3). If the vector space (subspace) consisting of the linear combination of these vectors is T, find the condition that the number of vectors constituting the basis of T will be 3.

The subspaces of R3 are planes, lines, R3 itself, or Z containing only (0, 0, 0).(a) Describe the three types of subspaces of R2 .(b) Describe all subspaces of D, the space of 2 by 2 diagonal matrices.

If W\index{1} and W\index{2} are subspace of a vector space V(F), then show that W\index{1}+W\index{2} is also a subspace of V(F).