[理工] [線代]-least squares相關問題

[ruby791104 (阿年：）)]

1.Find the least squares solution to each of the following systems.

  (a) x1 +  x2 = 3   (b)                 (c)

     2x1 - 3x2 = 1           BALABALA            BALABALA

     0x1 + 0x2 = 2

                             ^
2.For each of your solutions x in Excercise 1 ：

                                    ^
  (a) Determine the projection p = Ax.

                               ^
  (b) Calculate the residual r(x).

                    ^
  (c) Verify that r(x) € N(A^T).

3.For each of the following systems Ax = b, find all least squaressolutions.

          ╭      ╮     ╭ ╮
          │ 1  2 │     │3│
  (a)  A =│ 2  4 │, b =│2│ (b) BALABALA
          │-1 -2 │     │1│
          ╰      ╯     ╰ ╯

4.For each of the systems in Exercise 3, determine the projection p of b onto

  R(A) and verify that b-p is orthogonal to each of the column vectors of A.

嗯嗯……

我第一題和第三題都會算，

可是第二和第四題我不懂它是要求什麼＠＠

可以請大大講解順便示範一下嗎？

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◆ From: 120.127.32.231
推 AIdrifter:第2 (a)應該是問投影矩陣 然後(b)他問剩餘的向量是哪些
→ AIdrifter:N(A^T)=R(A)垂直 就是剩餘的向量 他要你證這個
→ AIdrifter:第四題要你算投影到A行空間的矩陣
→ AIdrifter:p是投影在A上 所以b-p垂直於A的行空間 大概是這樣
推 AIdrifter:少打 第四題要你算b投影到A的向量 就是p
→ ruby791104:謝謝AI大，我再去研究看看（鞠躬