## Multiplicity of an eigen value , k = dim[ Null(T - k I)^( dim V) ]

I have been reading Linear Algebra done right by Sheldon Axler
I got two conceptual queries :

(1) It states that for a Matrix of an operator T = [ 5 1 ; 0 5 ] ( ; indicates next row )
that dim Null [ (T - 5 I )2 ] = multiplicity of the eigen value 5 = 2

However, T - 5 I= [ 0 1 ; 0 0 ]

and (T - 5I )2 = [ 0 0 ; 0 0 ]

dim Null [ (T - 5 I )2 ] ≠ 2

I am a bit confused about the given result in the book hence. Could anyone please clarify.

(2) Multiplicity of an Eigen value , k = dim[ Null(T - k I)dim V ]

I have been trying to prove this without induction . Any direction please ?

thanks
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But the nullspace of $\left(\begin{array}{cc} 0 & 0\\ 0 & 0\end{array}\right)$ is $\mathbb{R}^2$ (all vectors are sent to 0). And the dimension of this is 2.