申論題

3.(20%) Let fx,y,z； be a set of linearly independent vectors in Rⁿ , and let S := Span(x,y) and T:= Span(y,z). Define matrix A:= xy^T+yz^T . Obviously, the  sets S, T, their orthogonal complements S^⊥, T^⊥, and the four sets associated with matrix A, ie. the two ranges R(A) and R(A^T) and the two null spaces N(A^T) and N(AT), are all subspaces of Rⁿ. 

This problem has three questions. The first one is a MULTIPLE-choice question, for which you don't need to give any derivation, but you need to give detailed derivations for the other two questions. In the multiple-choice question, the total score is evenly divided into each correct statement, and your each correct choice will get the partial score. However, the penalty for each wrong choice is equal to the score ofeach corectchoice.(所以同時選了一個對的答案和一個錯的答案時,淨得分為0 ；但是扣分僅扣到該小題0分為止。另外為方便改題、請將選擇題的答案寫在 此題做答處即可,不要寫到別處,以免漏改。)