---
# System prepended metadata

title: 研究所 線代

---

# 研究所 線代
## ==SVD==
![image](https://hackmd.io/_uploads/rJQSRinOyx.png)
![image](https://hackmd.io/_uploads/HyhrAi3_1x.png)
![image](https://hackmd.io/_uploads/S1qLRihOJg.png)
![image](https://hackmd.io/_uploads/SkEvRondke.png)

## ==ker(AHA)=ker(A)==
![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)
## ==rank(AHA)=rank(A)==
![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)

## ==A、B相似，rank(A)=rank(B)==
![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)

## ==A B克逆 => AB~BA==

![image](https://hackmd.io/_uploads/rJ1ZNborJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/rktJPHU8kl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==simultaneously diagonalizable==

![image](https://hackmd.io/_uploads/Hk7EnmHL1x.png)

![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==各類矩陣性質==
![image](https://hackmd.io/_uploads/SJH9CN8UJe.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/rkjh77HU1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/rkjXfmSIJe.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/S1ti8nzIJl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/HJtJP3fLJl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/rJBYvhM8yl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/r1qjD2MI1g.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==對稱矩陣性質==
![image](https://hackmd.io/_uploads/HkXIL2GL1l.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==伴隨算子==
![image](https://hackmd.io/_uploads/Hk4yu2fUJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==正規矩陣的性質==
![image](https://hackmd.io/_uploads/Bkyz_nGUke.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==正定矩陣的性質==
![image](https://hackmd.io/_uploads/r1AQu2fLJg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==正半定矩陣的性質==
![image](https://hackmd.io/_uploads/HyQHO2fLyg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==單式對角化條件==
![image](https://hackmd.io/_uploads/H1E8u3MUJl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==正交對角化條件==
![image](https://hackmd.io/_uploads/rkcuOnMLkx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==AT * A 性質==

![image](https://hackmd.io/_uploads/ByjYdhfLJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==正定矩陣分解==
![image](https://hackmd.io/_uploads/HJv2u2zLyg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)



## ==正半定矩陣分解==
![image](https://hackmd.io/_uploads/B1CA_2zL1l.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==判斷正定、正半定==

![image](https://hackmd.io/_uploads/r1U-YnMIye.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==要能正交對角化才能光譜分解==

![image](https://hackmd.io/_uploads/BJvVt2zU1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==任何矩陣都能QR、SVD分解==
![image](https://hackmd.io/_uploads/BkhvthMI1l.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)


## ==快速求eigen value與做對角化==
![image](https://hackmd.io/_uploads/Bkv5MwzLkx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)
![image](https://hackmd.io/_uploads/H143MPM8yx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)
![image](https://hackmd.io/_uploads/HkBRGwfIyl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==cholesky分解==
:::danger
1. 要是方陣
2. 要是對稱矩陣
3. 要是正定矩陣
:::
![image](https://hackmd.io/_uploads/S1xbinzI1g.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/S1doHUMUkl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/H13-8LM8yx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/HyD7UUM8yx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==相異eigen value對應的eigen vactor獨立==
:::info
若A是正規矩陣，則保證獨立且orthogonal
:::
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==只有實數對稱矩陣才可正交對角化==
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==LU==
![image](https://hackmd.io/_uploads/B1_KbnfE1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/BJPs-hMEkx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)
## ==manimal solution==
![image](https://hackmd.io/_uploads/BkrAjrhBye.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/S16yhHhrJg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/rJqxhr3HJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==orthogonal complement==
正交補空間

![image](https://hackmd.io/_uploads/H1lGVg3HJl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/B1Qarl3Hke.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==正交投影函數性質==
:::danger
對於所有向量 v屬於V，必存在唯一 (v1 屬於 S) 且 (v2 屬於 S perp)
滿足 v = v1+v2
:::
![image](https://hackmd.io/_uploads/Sk5xtH3Skl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/BJq3uHhSJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/Sk3FqBnrJl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==空間直和==

![image](https://hackmd.io/_uploads/SyZSoehBye.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)


![image](https://hackmd.io/_uploads/SkpWnehrkg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)
## ==行空間和空間互換==

![image](https://hackmd.io/_uploads/BJe4_b2Ske.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==獨立子空間==
![image](https://hackmd.io/_uploads/HJufqxhSkg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)


## ==正定、正半定等矩陣定義==
![image](https://hackmd.io/_uploads/SyL81ghN1g.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==rank判斷解的個數==

![image](https://hackmd.io/_uploads/B10rH7TE1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)
## ==相似det tr相同的證明==
![image](https://hackmd.io/_uploads/H1TaXWoHJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==for all等價==
:::danger
for all x => 對於所有正整數
exist x => 存在1正整數
p(x) =>為基數
q(x) =>維偶數
:::
![image](https://hackmd.io/_uploads/S1vI5JSB1l.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

# ==線性轉換==
:::danger
線性轉換要滿足
1. ![image](https://hackmd.io/_uploads/ry8DQafrJl.png)
2. ![image](https://hackmd.io/_uploads/Sy9um6Grkg.png)

:::

:::danger
1. 線性算子的矩陣表示法是方陣
2. T:v1->v2 ，dim(v1)=n,dim(v2)=m，***T的矩陣表示法是m*n**
:::
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==找A==
![image](https://hackmd.io/_uploads/rykE76GHye.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==旋轉矩陣==
![image](https://hackmd.io/_uploads/B1SBEpfr1l.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==線性轉換的矩陣表示法==

![image](https://hackmd.io/_uploads/SkqADcmByx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/B1y4v57Hyg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==換底==
![image](https://hackmd.io/_uploads/S1nCVi7H1g.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/H1P64iQryx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==換底重要題目==
![image](https://hackmd.io/_uploads/S1JudoXr1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/HJrKds7B1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/BJAt_o7S1g.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/HyI6usmB1g.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/HyIxtiQB1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==核集 值域==

![image](https://hackmd.io/_uploads/HyPPa1NHyg.png)

![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==求值域==

![image](https://hackmd.io/_uploads/BkMsnJVH1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)
## ==求核集==

:::danger
T(x,y)對應到0
:::

![image](https://hackmd.io/_uploads/ryKhny4Hyl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/H1EThkNBke.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/ByBC21ESyl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==維度定理==

![image](https://hackmd.io/_uploads/BJFF01EHJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/SJZfkxEHJl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/ByRz1x4r1e.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)


## ==injection bijection surjection==

![image](https://hackmd.io/_uploads/B1FAWgEB1e.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)


![image](https://hackmd.io/_uploads/B18GGx4Hye.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)


![image](https://hackmd.io/_uploads/S14zXeVrJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==維度相同時只要一對一或印成其中一個成立，另一個就會自動成立==

![image](https://hackmd.io/_uploads/Hk6Y7eNSJl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==任何線性轉換保持相依性==

![image](https://hackmd.io/_uploads/S1asExVSye.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==一對一保持獨立==

![image](https://hackmd.io/_uploads/HkjQUx4rJl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/SksSUlNr1g.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==onto保持生成==

![image](https://hackmd.io/_uploads/HytrDe4B1l.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/HkMDwl4Byx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==有解無解與rank的關西==
![image](https://hackmd.io/_uploads/r1RTDZNH1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/r1_26eEHke.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/BJUC6l4ryx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)


# ==idempotent matrix==
![image](https://hackmd.io/_uploads/HkbPKVVB1e.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/BkXD0NVHJx.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/Byn2RENBkl.png)

![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==sylvester 2 law==
![image](https://hackmd.io/_uploads/BkiB0VNr1l.png)

![image](https://hackmd.io/_uploads/Skbwyg241x.png)
## AHA 與 A的關西
![image](https://hackmd.io/_uploads/H1lDgSVB1x.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)
![image](https://hackmd.io/_uploads/BykClBNHke.png)
![image](https://hackmd.io/_uploads/H1lDgSVB1x.png)

# ==最小平方解==
:::danger
AT*A*X=AT*B 定有解
1. 當A行獨立 => AT*A可逆 => 唯一解
2. 當A行不獨立 => 無限多解
:::
![image](https://hackmd.io/_uploads/HyAkNW4HJl.png)
![image](https://hackmd.io/_uploads/SJ_eVWNHyg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

## ==正交投影==
:::danger
要先確定basis是正交的 => 作正交化
:::
![image](https://hackmd.io/_uploads/Hyw9SNErkg.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)

![image](https://hackmd.io/_uploads/H1pamHVHyl.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)
## ==求回歸線==
![image](https://hackmd.io/_uploads/HktBVBEH1l.png)
![image](https://hackmd.io/_uploads/Skbwyg241x.png)


