Is Buv a subspace Why or why not Is S a subspace Why or why

Is B={u,v} a subspace? Why or why not?

Is S a subspace? Why or why not?

Given u = [-3 5 0 0] v = [0 5 -1 4] and S = span {u, v}. Is B = {u, v} a subspace? Why or why not? Is S a subspace? Why or why not?

Solution

A linear subspace (or vector subspace) is a vector space that is a subset of some other (higher-dimension) vector space. A linear subspace is usually called simply a subspace. The vectors u, v are linearly independent as can be observer from reducing to the RREF form, the matrix with u and v as columns. Then, B = {u,v} forms a basis for S.

a. B is not a subspace in itself as otherwise, it would be closed under vector addition and scalar multiplication. Also then, the zero vector has to be in B. Since these three conditions are not satisfied, B is not a subspace.

b. S is a subspace of R⁴ as by definition of span {u, v}, any linear combination of u and v is in S and therefore S is closed under vector addition. Also any scalar multiple of both u and v is in S( by definition itself). When this scala is 0, the zero vector also is in S. Therefore S is a sub-space.