91影视

Let V be a vector space over a field K. Let $v_1,v_2,v_3,...,v_n鈭�V$if there exist scalars $a_1,a_2,a_3,...,a_n$K, not all of them 0, such that

$a_1{v}_1+a_2{v}_{+}...a_n{v}_n=0$

Then the vectors$v_1,v_2,v_3,...,v_n$ are called linearly dependent vectors.

Otherwise, linearly independent vectors.

A matrix A of order n x n is said to be nilpotent if there exists a smallest positive integer 鈥榤鈥� for which

$A^m=0but{A}^{m-1}鈮�0$

Then the smallest positive integer 鈥榤鈥� is called an index of the nilpotent matrix.

Let us consider a nilpotent matrix A of order 3 x 3 such that

$A^2=0butA鈮�0$

Then we have a nilpotent matrix A of index 2 and so 鈥榤鈥� = 2.

Considering the vectors $\stackrel{鈫�}{v},A\stackrel{鈫�}{v},A^2\stackrel{鈫�}{v},...,A^{m-1}\stackrel{鈫�}{v}$are linearly dependent then, there exist scalars $a_1,a_2,a_3,鈥�,a_m$, not all of them 0, such that

$a_1\stackrel{鈫�}{v}+a_2\stackrel{鈫�}{v}鈨�+a_3{A}^2\stackrel{鈫�}{v}+...+a_m{A}^{m-1}\stackrel{鈫�}{v}=0$

Since we have considered a nilpotent matrix A of order 3 x 3 such that

$A^2=0butA鈮�0$

where 鈥榤鈥� = 2, then we have to show that the vectors are linearly independent.

Let us suppose the vectors $\stackrel{鈫�}{v}$and $A\stackrel{鈫�}{v}$are linearly dependent, then there exist scalars a and b, not all of them 0, such that

$a\stackrel{鈫�}{v}+{}_{b}A\stackrel{鈫�}{v}=0$ (1)

Multiply equation (1) by A we get,

$aA\stackrel{鈫�}{v}+b{A}^2\stackrel{鈫�}{v}=0$ (2)

$$\begin{gathered} 鈭�A^2=0 \\ 鈬�b{A}^2\stackrel{鈫�}{v}=0(鈭�A\stackrel{鈫�}{v}鈮�0) \end{gathered}$$

Thus from equation (2), we have

$$\begin{gathered} aA\stackrel{鈫�}{v}=0 \\ 鈬�a=0 \end{gathered}$$

Therefore from equation (1), we have

$bA\stackrel{鈫�}{v}=0$

$鈬�b=0(鈭�A\stackrel{鈫�}{v}鈮�0)$

Thus, for $a\stackrel{鈫�}{v}+{}_{b}A\stackrel{鈫�}{v}=0$we get $a=0andb=0$.

Hence, our supposition that the vectors$\stackrel{鈫�}{v}$and$A\stackrel{鈫�}{v}$ are linearly dependent is wrong and so the vectors$\stackrel{鈫�}{\mathrm{v}}\mathrm{and}\mathrm{A}\stackrel{鈫�}{\mathrm{v}}$ are linearly independent.

Since it is true for a 3 x 3 matrix; hence it is true for n x n matrix A such that

$A^m=0but{A}^{m-1}鈮�0$

Thus the vectors $\stackrel{鈫�}{v},A\stackrel{鈫�}{v},A^2\stackrel{鈫�}{v},...,A^{m-1}\stackrel{鈫�}{v}$are linearly independent.

Let $m鈮�n$. Then

$$\begin{gathered} A^n=A^n{A}^m{A}^{-m}=A^{n-m}{A}^m=0(鈭�A^m=0) \\ 鈬�A^n=0 \end{gathered}$$

Now, let $m>n$.

This implies the number of linearly independent vectors is greater than 鈥榤鈥� which is a contradiction as the number of linearly independent vectors in an n x n matrix is m.

Thus,$m鈮�n$

If we take a nilpotent 3 脳 3 matrix A and choose the smallest number 鈥榤鈥� = 2 such that $A^2=0butA鈮�0$and pick a vector $\stackrel{鈫�}{v}$in ${\mathrm{鈩�}}^n$such that $A\stackrel{鈫�}{v}鈮�0$then the vectors$\stackrel{鈫�}{v}$are$A\stackrel{鈫�}{v}$ linearly independent.

Since it is true for the 3 x 3 matrix; hence it is true for n x n matrix A such that

$A^m=0but{A}^{m-1}鈮�0$

Thus the vectors$\stackrel{鈫�}{v},A\stackrel{鈫�}{v},A^2\stackrel{鈫�}{v},...,A^{m-1}\stackrel{鈫�}{v}$ are linearly independent.

If$m鈮�n鈬�A^n=0$