Question

Let U, V be subspaces of R^n. Prove that U + V = {u + v : u ∈ U and v ∈ V} is also a subspace of R^n.

Answer 1

Clearly ,U+V is non empty and zero vector is belongs to this set because 0 belongs to U and V as they are Subspace. Now let

$u_1 ,u_2 \in U \, and \space v_1 ,v_2 \in V .\,\\ Then \space u_1+v_1 ,u_2+v_2 \in U+V.\,\\ Now \space u_1+v_1+u_2+v_2=(u_1+u_2)+(v_1+v_2)\in U+V.$$Since \space U \, and \space V \, are \, subspaces$ .

Let $k \in K$ ,where K is the given field.

Therefore,$k(u_1+u_2)=ku_1+ku_2 \in U+V.$

Since U and V are subspace.

Hence U+V is a subspace of $R^n$ .