Vector Spaces - Linear Algebra

Question

Let R and C be the field of real numbers and the field of complex numbers, respectively. Let M_m,n(R) be the set of all m × n matrices over R and let M_n(R) = M_n,n(R).

Let R 3 = {(x, y, z) : x, y, z ∈ R} be the vector space of 3-tuples of real numbers under the usual addition and scalar multiplication. Determine which of the of the following are subspaces of R 3 .

(a) S1 = {(a, b, c2 ) : a, b, c ∈ R}.

(b) S2 = {(a, b, c) ∈ R 3 : ax + cy = 0 for all x, y ∈ R}.

Justify your answers.

Answer 1

a) This is not a subspace of R^3 since the vector u=(0,0,1) belongs in S_1 but it has no opposite, i.e. there is no vector v in S_1 such that u+v=(0,0,0).

b) This is a subset of R^3. It simply follows by checking the definition.