Linear Algebra lecture8 note

標籤：pad   無法   線性   png   exist   display   技術分享   complete   .com   

Compute solution of AX=b (X=Xp+Xn)

rank r

r=m solutions exist

r=n solutions unique

 

 

example:

若想方程有解，b1,b2,b3需要滿足什麼條件？ 觀察矩陣可知，第三行是前兩行的和，所以b1+b2=b3

Solvability Condition on b:

Ax=b is solvable when b is in C (A)

If a combination of Rows of A gives zero row, then the same combination of entries of b must give 0

假設，則上述矩陣變為：

To find complete solution to AX=b:

1.Xp (particular): set all free variables to zero, solve AX=b for pivot variable

此例中，X2=0，X4=0

2.Xn(nullspace) 上一節已經解出

3.X(complete)=Xp+Xn

以上操作可解釋為：

 

 

m by n matrix A of rank r(r<=m,r<=n)

Full column of rank(r=n):

所有列均有主元; no free variables;  N(A)=zero vector; solution to AX=b is X=Xp which means if solution exists then the solution is unique(0 or 1 solution)

這種情況實際就是，除zero組合之外，列之間的線性組合無法產生零列

Full row of rank(r=m):

所有行均有主元; no zero rows; can solve AX=b for every b; left with n-r(n-m) free variables

Full rank(r=m=n):

N(A)=zero vector; R(行最簡形）=I（單位矩陣）

 

summary：

矩陣的秩決定了方程組解的數目

Linear Algebra lecture8 note