10/28 作业

Homework 12

1. (1) $\left\{ (1, 1, 1), (1, 2, 3) \right\}$ , $\left\{ (-2, 1, 0) \right\}$ .

   (2) $\left\{ (0, 1), (1, 0) \right\}$ , $\left\{ (-2, 1, 0, 0, 0), (0, 0, 2, -5, 3), (0, 3, 8, -2, 0) \right\}$ .

   (3) $\left\{ (1, 0, 0, 0), (1, 1, 1, 1), (1, 2, 3, 4) \right\}$ , $\left\{ (-3, -2, 1, 0) \right\}$ .

   (4) $\left\{ (-3, 1, 2), (-1, 2, 5) \right\}$ , $\left\{ (-10, -5, 0, 0, 0), (0, 0, -20, 15, 5), (0, 5, 20, -10, 0) \right\}$ .

   (5) $\left\{ (-4, -3, 2, 5), (2, 4, 6, 0), (5, 7, 9, -9) \right\}$ , $\left\{ (0, -22, -34, 6, 4), (4, -16, -24, 0, 0) \right\}$ .

2. (a) No.

   (b) No.

   (c) Yes.

   (d) No.

3. $\R - \left\{ 29 \right\}$ .

4. $\left\{ (1, 2, 0, 0), (0, 2, 1, 0), (0, 4, 0, 1) \right\}$ .

5. $\dim V \ge m - n$ .

   Proof Let $A$ be the coefficient matrix. Then $\dim V = \operatorname{nullity} A$ . Since $\operatorname{rank} A + \operatorname{nullity} A = m$ , and $\operatorname{rank} A \le n$ , we have $\dim V \ge m - n$ $\square$ .

6. (1) $L$ .

   (2) $X$ .

   (3) $H$ , $L$ .

7. (a) False.

   (b) False.

   (c) True.

   (d) False.

8. a. Equal.

   b. Equal.