Linear algebra study notes 4-4: Solving the inhomogeneous system of linear equations Ax=b, looking at the equations from the perspective of rank

$\mathbf{A}\mathbf{x}=\mathbf{b}$与$\mathbf{A}\mathbf{x}=0$的区别在于：

• 对于齐次线性方程组$\mathbf{A}\mathbf{x}=0$,No matter how the left coefficient matrix is ​​row transformed,The coefficient on the right is always$0$;
  $\mathbf{A}\mathbf{x}=0$必定有解,It's just that the solution space may be large or small（唯一零解？无穷解？）
• 然而对于$\mathbf{A}\mathbf{x}=\mathbf{b}$,消元时,右侧的$\mathbf{b}$and the left-hand coefficient matrix$\mathbf{A}$Transform together,We need to consider augmented matrices$\left[\begin{array}{cc}\mathbf{A}& \mathbf{b}\end{array}\right]$
  $\mathbf{A}\mathbf{x}=\mathbf{b}$可能无解,there may also be a solution;A solution may be the only solution/无穷解

$\mathbf{A}\mathbf{x}=\mathbf{b}$有解,一定要保证：Once the linear combination of some lines on the left is the full0,The coefficient on the right side must also be0（After elimination, the equation always satisfies"0=0"的约束）
例如,$\left[\begin{array}{cc}\mathbf{A}& \mathbf{b}\end{array}\right]=\left[\begin{array}{ccccc}1& 2& 2& 2& b_1\\ 2& 4& 6& 8& b_2\\ 3& 6& 8& 10& b_3\end{array}\right]$,get after elimination$\left[\begin{array}{ccccc}1& 2& 2& 2& b_1\\ 0& 0& 2& 4& b_2-2b_1\\ 0& 0& 0& 0& b_3-b_2-b_1\end{array}\right]$,则必须满足$b_3-b_2-b_1=0$

$\mathbf{A}\mathbf{x}=\mathbf{b}$有解的条件

• for augmented matrices$\left[\begin{array}{cc}\mathbf{A}& \mathbf{b}\end{array}\right]$,若$\mathbf{A}$The linear combination of certain lines yields the full0行,对应的$\mathbf{b}$side is also0,则方程$\mathbf{A}\mathbf{x}=\mathbf{b}$有解

Why is it so certain that the equation has a solution?？
First of all this is saying,There is no between the equations“互相矛盾”（Didn't show up after disappearing"0=1"）
其次,We start solving from the last pivot,Bring back up,The unknowns must be solved according to the constraints of the equation,But it is not certain how many solutions there are.

• 仅当$\mathbf{A}$的列空间$C(\mathbf{A})$包含向量$\mathbf{b}$时,方程$\mathbf{A}\mathbf{x}=\mathbf{b}$有解
  [矩阵乘法角度]这就是说,A linear combination of column vectors can be obtained$\mathbf{b}$
  [geometric angle]矩阵$\mathbf{A}$Linearly transformed space（base is a column vector）,包含$\mathbf{b}$
  因此,We can first find the column space,然后判断是否有解

求解$\mathbf{A}\mathbf{x}=\mathbf{b}$

求解$\mathbf{A}\mathbf{x}=\mathbf{b}$,$\mathbf{A}\mathbf{x}=\mathbf{b}$的一个特解 + $\mathbf{A}\mathbf{x}=0$all possible solutions of = $\mathbf{A}\mathbf{x}=\mathbf{b}$all possible solutions of

具体步骤：$\mathbf{A}\mathbf{x}=\mathbf{b}$使用5-3The elimination method in question gets a special solution,$\mathbf{A}\mathbf{x}=0$Use the elimination method to obtain the basic solution system and null space,The two can be superimposed

• 原因：$x_p$为$\mathbf{A}\mathbf{x}=\mathbf{b}$的特解,$x_n$为$\mathbf{A}\mathbf{x}=0$all possible solutions of（$\mathbf{A}$all vectors in the null space of）,那么有$\mathbf{A}{\mathbf{x}}_p=\mathbf{b}$和$\mathbf{A}{\mathbf{x}}_n=0$,两个方程相加,得到$\mathbf{A}({\mathbf{x}}_{\mathbf{p}}+{\mathbf{x}}_{\mathbf{n}})=\mathbf{b}$,从而所有$({\mathbf{x}}_{\mathbf{p}}+{\mathbf{x}}_{\mathbf{n}})$都是方程$\mathbf{A}\mathbf{x}=\mathbf{b}$的解
• 注意,$\mathbf{A}\mathbf{x}=\mathbf{b}$的解空间一定不是向量空间,因为不含$0$向量,This solution space can be understood as：$\mathbf{A}\mathbf{x}=0$的零空间（向量空间）and equation particular solution vector$x_p$的叠加（Zero space for translation,好比y=x到y=x+b的平移）
• its implied meaning：
  ①若$Rank(\mathbf{A})=n$,那么$\mathbf{A}\mathbf{x}=0$有唯一零解,则$\mathbf{A}\mathbf{x}=\mathbf{b}$only solution（如果有解）
  ②若$Rank(\mathbf{A})<n$,Then the matrix corresponds to the dimensionality reduction transformation,那么$\mathbf{A}\mathbf{x}=0$solution space（dimension greater than zero,An infinite a non-zero solution）,进而$\mathbf{A}\mathbf{x}=\mathbf{b}$的通解=$\mathbf{A}\mathbf{x}=\mathbf{b}$的一个特解 + $\mathbf{A}\mathbf{x}=0$any linear combination of the fundamental solution system of

From the perspective of rank equation

定义秩$Rank(\mathbf{A})$：$m$行$n$列的系数矩阵$\mathbf{A}$After elimination,number of pivots,and must have$r\le m$和$r\le n$

• If a column full rank$Rank(\mathbf{A})=n<m$,那么$\mathbf{A}\mathbf{x}=0$有唯一零解,则$\mathbf{A}\mathbf{x}=\mathbf{b}$only solution（如果有解）
  原因：这意味着方程$\mathbf{A}\mathbf{x}=0$Each column has a pivot,All variables are the main variables,Always be on the right side and equations0,So get the only solution$0$,That is, the null space is a point,进一步导致$\mathbf{A}\mathbf{x}=\mathbf{b}$无解/唯一解
• If the row is full$Rank(\mathbf{A})=m<n$,那么对于任意$\mathbf{b}$,$\mathbf{A}\mathbf{x}=\mathbf{b}$必定有解,且有$n-Rank(\mathbf{A})$个自由变量（This is also the number of vectors in the basic solution system/零空间的维数）
  原因：This means that after the elimination is complete,左侧的$\mathbf{A}$did not appear in full0行,Therefore, there is no situation where the equations contradict each other.（如"0=1"）
• 行列满秩/满秩$Rank(\mathbf{A})=n=m$,$\mathbf{A}\mathbf{x}=0$唯一零解（Null space is a point）,$\mathbf{A}\mathbf{x}=\mathbf{b}$有唯一解
  理解：After elimination, each row and column has pivot,Then the row least echelon form must be the identity matrix
• 若$Rank(\mathbf{A})<n且Rank(\mathbf{A})<m$,Then the matrix corresponds to the dimensionality reduction transformation,那么$\mathbf{A}\mathbf{x}=0$有无穷个解,进而$\mathbf{A}\mathbf{x}=\mathbf{b}$有无穷个解

Reduced echelon form from row after elimination$\mathbf{R}$来看：

• Does the equation have a solution,就要看$\mathbf{R}$Does all appear0行
  If there is a whole0行,右边的$\mathbf{b}$对应为0,才有解;否则无解
• Does the equation have a unique solution or an infinite solution,就要看$\mathbf{R}$The number of free variables in $n-r$（列数-秩/主元个数）
  0free variables, the equation has a unique solution;1more than one free variable,方程有无穷解（How many free variables,Several basic solution system of vector,Thus zero space dimension is few）

specific to each situation：

• $Rank(\mathbf{A})=m=n$,行简化阶梯型$\mathbf{R}=\mathbf{I}$,not complete0行,Equation must have a solution,and the number of free variables is$0$,有唯一解
• $Rank(\mathbf{A})=n<m$,$\mathbf{R}=\left[\begin{array}{c}\mathbf{I}\\ 0\end{array}\right]$,when the corresponding row$\mathbf{b}$为0时方程才有解;若有解,The number of free variables is$0$,唯一解
• $Rank(\mathbf{A})=m<n$,$\mathbf{R}=\left[\begin{array}{cc}\mathbf{I}& \mathbf{F}\end{array}\right]$（After rearranging the variables you get this form）,not complete0行,Equation must have a solution,The number of free variables is$n-r>0$,有无穷解（Zero space, at least, is a straight line）
• $Rank(\mathbf{A})<n且Rank(\mathbf{A})<m$,$\mathbf{R}=\left[\begin{array}{cc}\mathbf{I}& \mathbf{F}\\ 0& 0\end{array}\right]$,when the corresponding row$\mathbf{b}$为0时方程才有解;若有解,The number of free variables is$n-r>0$,有无穷解

总之,
$\mathbf{A}$的零空间维数$dim[N(\mathbf{A})]$=$\mathbf{A}$After the elimination of free column number $n-Rank(\mathbf{A})$
秩$Rank(\mathbf{A})$=矩阵$\mathbf{A}$the number of pivot columns=$\mathbf{A}$Maximum number of linearly independent column vectors=$\mathbf{A}$the column space dimension of$dim[C(\mathbf{A})]$
秩$Rank(\mathbf{A})$determines the structure of the solution to the equation（有没有解、有唯一/无穷解）