1 1 2 2.1 m × n m × 1 = [w 1 w 2 . . . w n ] T n × 1 w j ≥ 0, ∀j = 1, , n m × n = [r 1 r 2 . . . r n ] T n × 1 r j > 0, ∀j = 1, , n W 1 , W 2 , . . . , W n {W 1 , W 2 , . . . , W n } rank(W) ≤ min{m, n − 1}. 2.2 R k { −→ v 1 , −→ v 2 , . . . , −→ v s } −−→ v s+1 rank{ −→ v 1 , −→ v 2 , . . . , −−→ v s+1 } = rank{ −→ v 1 , −→ v 2 , . . . , −→ v s } + 1, −−→ v s+1  −→ v 1 , −→ v 2 , . . . , −→ v s  { −→ v 1 , −→ v 2 , . . . , −→ v s } 2.3 ∗ = [r 1 r 2 . . . r n ] T n × 1 r j > 0, ∀j = 1, , n m × 1 ∗ W 1 =  0 0 I I  ; W 2 =  −1 1 1 −1  . 3 3.1 m × n m < n W 1 , W 2 , . . . , W n R m m < n m > n rank{W 1 , W 2 , . . . , W n } < m t ∈ R m rank{W 1 , W 2 , . . . , W n , t} = rank{W 1 , W 2 , . . . , W n } + 1, W 1 , W 2 , . . . , W n  {W 1 , W 2 , . . . , W n } m = n rank(W) < m m = n rank(W) = m {W 1 , W 2 , . . . , W n } R m = [r 1 r 2 . . . r n ] T n × 1 r j > 0, ∀j = 1, , n r 1 W 1 + r 2 W 2 + · · · + r n W n = 0. {W 1 , W 2 , . . . , W n } m rank{W 1 , W 2 , . . . , W n } < m t ∈ R m rank{W 1 , W 2 , . . . , W n , t} = rank{W 1 , W 2 , . . . , W n } + 1, W 1 , W 2 , . . . , W n  {W 1 , W 2 , . . . , W n } 3.2 m × n n > 1 k ≤ min{m, n − 1} m × n rank(W) = k ∗ n = 1 R m ∗ n > 1 W 1 , W 2 , . . . , W n−1 ∈ R m k r j > 0, ∀j = 1, , n W n = − r 1 r n W 1 − r 2 r n W 2 − · · · − r n−1 r n W n−1 ∈ R m (1) ⇒ r 1 W 1 + r 2 W 2 + · · · + r n W n = 0 (2) W 1 , W 2 , . . . , W n Wr = 0 m × n W 1 , W 2 , . . . , W n−1 ∈ R m k W n W 1 , W 2 , . . . , W n − 1 rank(W) = k m × n k 3.3 m × n (m < n) = [r 1 r 2 . . . r n ] T n × 1 r j ≥ 0, ∀j = 1, , n 0 r j > 0 R m W 1 , W 2 , . . . , W n W 1 , W 2 , . . . , W m r 1 > 0, r 2 > 0, . . . , r m > 0 r 1 W 1 + r 2 W 2 + · · · + r n W n = 0 (3) ⇔ r 1 W 1 + r 2 W 2 + · · · + r m W m = −r m+1 W m+1 − r m+2 W m+2 − · · · − r n W n . (4) m × 1 {W 1 , W 2 , . . . , W m } t = t 1 W 1 + t 2 W 2 + · · · + t m W m . (5) ∗ t j ≥ 0, j = 1, , m  w j = t j , j = 1, , m w j = 0, j = m + 1, , n (6) ⇒ t = w 1 W 1 + w 2 W 2 + · · · + w n W n , (w j ≥ 0, j = 1, , n) ∗ ∃t j < 0 r j > 0, j = 1, , m min{ t j r j | t j < 0, j = 1, m} min{ t j r j | t j < 0, j = 1, , m = t m r m . (7) t m < 0, r m > 0 t m r m < 0. w j = t j − t m r m .r j , (j = 1, , m) (8) ⇔ w j = r j  t j r j − t m r m  , (j = 1, , m) . (9) t j < 0 t j r j ≥ t m r m ⇔ t j r j − t m r m ≥ 0 ⇒ w j ≥ 0. t j > 0 t m r m < 0 w j = t j − t m r m .r j ≥ 0. w j ≥ 0, j = 1, , m. (10) t j = w j + t m r m .r j , j = 1, , m. (11) t j t = m  j=1 t j .W j = m  j=1  w j + t m r m .r j  W j = m−1  j=1  w j + t m r m .r j  W j + t m .W m = m−1  j=1 w j .W j + m−1  j=1  t m r m .r j  W j + t m r m r m .W m = m−1  j=1 w j .W j + m  j=1 t m r m r j .W j = m−1  j=1 w j .W j + t m r m  m  j=1 r j .W j  = m−1  j=1 w j .W j + t m r m  − n  j=m+1 r j .W j  = m−1  j=1 w j .W j + n  j=m+1  − t m r m r j  .W j . w j = − t m r m .r j , j = 1, , n (12) t m < 0, r m > 0, r j ≥ 0 w j ≥ 0, j = m + 1, , n. (13) w j ?? w m = 0 t = n  j=1 w j .W j (w j ≥ 0, j = 1, , n) , 3.4 m × n (m < n) ∗ ∗ n × 2n n = [1 1 . . . 1] T 3.5 m × n m < n ∗ ∗ Nhn xt. k = m 3.6 min{f(x)}  Ax = b x ≥ 0, (14) [x 1 x 2 . . . x n ] T n × 1 [b 1 b 2 . . . b m ] T m × 1 [a ij ] m × n f 3.7 m × n m ≥ n m < n {x | Ax = b, x ≥ 0} (14) {x | Ax = b, x ≥ 0} m × n m < n m 3.8 m × n α m × 1 [w 1 w 2 . . . w n ] T α w j ≥ α, j = 1, , n α m × 1 t ∗ = t − Wα ∗ α ∗ = [α α . . . α] T n × 1 [y 1 y 2 . . . y n ] T y j ≥ 0, j = 1 , , n t ∗ Wα ∗ W(y + α ∗ ) = t α ∗ = [w 1 w 2 . . . w n ] T w j = y j + α ≥ α α α = 0 3.9 m × n α m × 1 [w 1 w 2 . . . w n ] T α w j ≤ α, j = 1, , n α m × 1 t ∗ = −t + Wα ∗ α ∗ = [α α . . . α] T n × 1 [y 1 y 2 . . . y n ] T y j ≥ 0, j = 1, , n t ∗ Wα ∗ W(−y + α ∗ ) = t α ∗ = [w 1 w 2 . . . w n ] T w j = −y j + α ≤ α α α = 0 m × 1 t ∗ = −t [x 1 x 2 . . . x n ] T 0 x j ≤ 0, j = 1, , n t ∗ [w 1 w 2 . . . w n ] T w j = −x j ≥ 0, j = 1, , n 56 2