Suppose S and T are two subspaces of a vector space V.(a) Definition: The sum S + T contains all sums s + t of a vectors in S and a vector tin T. Show that S + T satisfies the requirements (addition and scalar multiplication) for a vector space.(b) If S and T are lines in Rm , what is the difference between S + T and S U T? That union contains all vectors from S or T or both. Explain this statement: The span of SU T is  S + T. (Section 3.5 returns to this word "span".)

Can you explain why the statement in the image is true?

Suppose that V is a that has subspaces of U and W. Furthermore, suppose that {u¹, u2} is a basis for U, that {1, 2} is a basis for W and that the only vector that U and W have in common is the zero vector 0. Show that {u¹, u², w¹, w²}. After you get done, note where you used the fact that the only vector common to both U and W is 0. (Without this condition the vectors {u¹, u², w², w2} don't need to be linearly independent, so if you didn't use this condition you definitely made a mistake somewhere.)

Please provide a solution using the answer key I have attached below.