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Problem:State True or False


The set of nilpotent matrices of ( M_3 (\mathbb{R} ) ) spans ( M_3 (\mathbb{R} ) ) considered as an ( \mathbb {R} ) – vector space ( a matrix A is said to be nilpotent if there exists ( n \in \mathbb{N} )  such that ( A^n = 0 ) ).


Discussion


If such a basis exist. Then it will contain 9 matrices (because dimension of ( M_3 ( \mathbb{R} ) ) is 9 ).

Suppose ( { N_1, … , N_9 } ) be the 9 nilpotent matrices which span ( M_3 (\mathbb{R} ) ). Now consider the usual basis of ( M_3 (\mathbb{R} ) ) :

$$ M_1 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} , M_2 = \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} , … , M_9 = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}$$

There is a linear transformation L (via a ( 9 \times 9 )  change of basis matrix) that sends ( { N_1, … , N_9 } ) to ( { M_1, … , M_9 } ).


Chatuspathi

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