91影视

For a linear system $\mathit{A}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{=}\mathit{b}$, a vector ${\stackrel{\mathbf{鈬赌}}{\mathbf{x}}}^{\mathbf{*}}$in localid="1662034423145">鈩�nis considered as least-squares solution of the given linear system if$||\stackrel{鈬赌}{b}-A{\stackrel{鈬赌}{x}}^{*}||\mathbf{鈮�}||\stackrel{鈬赌}{b}=A\stackrel{鈬赌}{x}||$for every$\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$in localid="1662034439414">鈩�m.

Suppose there is a consistent system $A\stackrel{鈬赌}{x}=\stackrel{鈬赌}{b}$. There exists a unique solution ${\stackrel{鈬赌}{x}}_0鈭�{\left(kerA\right)}^{鈯�}$.

Now, it is sure that there is another solution of the system, that is ${\stackrel{鈬赌}{x}}_1$, if ${\stackrel{鈬赌}{x}}_0鈭�{\left(kerA\right)}^{鈯�}$. The above statement implies that$||{\stackrel{鈬赌}{x}}_0||鈮�||{\stackrel{鈬赌}{x}}_1||$ which means the minimal solution of the linear system is ${\stackrel{鈬赌}{x}}_0$.

It is known that the least-squares solution are the solutions of system$A^{T}A\stackrel{鈬赌}{x}=A^T\stackrel{鈬赌}{b}$ for any linear system $A\stackrel{鈬赌}{x}=\stackrel{鈬赌}{b}$.

From the above statements, the minimal least-square solution of a linear system $A\stackrel{鈬赌}{x}=\stackrel{鈬赌}{b}$is the minimal solution of the system role="math" localid="1660661580509" $A^{T}A\stackrel{鈬赌}{x}=A^T\stackrel{鈬赌}{b}$.

Now, role="math" localid="1660661594123" ${\stackrel{鈬赌}{x}}^{*}$is the minimal solution of the system $A^{T}A\stackrel{鈬赌}{x}=A^T\stackrel{鈬赌}{b}$. So, ${\stackrel{鈬赌}{x}}^{*}$is the minimal least-squares solution of the consistent system $A\stackrel{鈬赌}{x}=pro{j}_{im\left(A\right)}\stackrel{鈬赌}{b}$ is in im(A). Since, we know that $im\left(A\right)={\left(kerA\right)}^{鈯�}$.

Thus, it is proved that ${\stackrel{鈬赌}{x}}^{*}鈭�{\left(kerA\right)}^{鈯�}$.