91影视

Let A be a matrix from ${\mathrm{鈩�}}^n\text{to}{\mathrm{鈩�}}^m鈬�\stackrel{鈫�}{u}=0$then the kernel of matrix A, denoted by ker (A), is defined as-

$\ker \left(A\right)=\left\{x鈭�{\mathrm{鈩�}}^n:Ax=0\right\}$

And the image of the matrix A, denoted by Im (A), is defined as-

$Im\left(A\right)=\left\{Ax:x鈭�{\mathrm{鈩�}}^n\right\}$

We have given a n x n matrix A such that $A^2=A$.

Let us consider a vector $\stackrel{鈫�}{u}鈭�\ker \left(A\right)鈭�Im\left(A\right)$then by definition of the image of A and kernel of A we have $\stackrel{鈫�}{u}=A\stackrel{鈫�}{w}$and $\stackrel{鈫�}{u}=\stackrel{鈫�}{0}$ for some vector$w鈭�{\mathrm{鈩�}}^n.$

$鈬�A\stackrel{鈫�}{w}=0$

Now multiplying the above equation by A from left, we get

role="math" localid="1664174647393" $$\begin{gathered} A\stackrel{鈫�}{u}=A^2\stackrel{鈫�}{w} \\ 鈬�A\stackrel{鈫�}{u}=A\stackrel{鈫�}{w}=0\left(鈭�A^2=A,A\stackrel{鈫�}{w}=0\right) \end{gathered}$$

role="math" localid="1664174670751" $鈬�A\stackrel{鈫�}{u}=\stackrel{鈫�}{u}=0\left(鈭�A\stackrel{鈫�}{w}=\stackrel{鈫�}{u}\right)$

$鈬�\stackrel{鈫�}{u}=0$

$鈬�$The only vector common to both image of A and the kernel of A is the zero-vector.

If A is any n 脳 n matrix such that$A^2=A$ , then any vector$\stackrel{鈫�}{u}鈭�\ker \left(A\right)鈭�Im\left(A\right)$ .

$鈬�\stackrel{鈫�}{u}=0$

Hence, the image of A and the kernel of A have only the zero vector in common.