Consider the multiplication operator $L_A:\mathbb{R}^4\to\mathbb{R}^4$ defined by $L_A(x)=Ax$, where $A=\left[\begin{array}{cccc} -41 &55 &-60 &-17\cr -6 &8 &-9 &-2\cr 26 &-35 &38 &11\cr -14 &19 &-21 &-5 \end{array}\right]$.
a. Find the smallest nonnegative $k$ such that the ranks of $L_A^k$ and $L_A^{k+1}$ are the same.
$k=$
It can shown that $\mathbb{R}^4=\ker(L_A^k)\oplus\hbox{ran}(L_A^k)$. This means $\ker(L_A^k)\cap\hbox{ran}(L_A^k)=\lbrace \underline{0} \}$ and $\mathbb{R}^4=\ker(L_A^k)+\hbox{ran}(L_A^k)$.
b. Find a matrix $K$ whose row space is $\ker(L_A^k)$.
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c. Find a matrix $R$ whose row space is $\hbox{ran}(L_A^k)$.
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d. Find a vector $u\in \ker(L_A^k)$ and a vector $v\in \hbox{ran}(L_A^k)$ such that $u+v = \left(-9,0,5,4\right)$
$u=$ , $v=$