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exposition/problem 2.tex b/research/04 Parigots exposition/problem 2.tex deleted file mode 100644 index 45c45bd9..00000000 --- a/research/04 Parigots exposition/problem 2.tex +++ /dev/null @@ -1,73 +0,0 @@ -\documentclass{amsart} - -\usepackage{../AMC_style} -\usepackage{../Research} - -\DeclareMathOperator{\TT}{\boldface T} -\DeclareMathOperator{\A}{\boldface A} -\DeclareMathOperator{\B}{\boldface B} -\DeclareMathOperator{\PR}{P} - -\begin{document} - - \begin{enumerate} - \item Show that if $A \sim B$ then $A^n \sim B^n$ - \item Show that - \begin{align*} - \begin{bmatrix} - 0 & 1 & 0 \\ - 0 & 0 & 1 \\ - 0 & 0 & 0 - \end{bmatrix} - \nsim - \begin{bmatrix} - 0 & 1 & 0 \\ - 0 & 0 & 0 \\ - 0 & 0 & 0 - \end{bmatrix} - \end{align*} - \item Show that - \begin{align*} - \begin{bmatrix} - 0 & 0 & 0 \\ - 0 & 0 & 1 \\ - 0 & 0 & 0 - \end{bmatrix} - \sim - \begin{bmatrix} - 0 & 1 & 0 \\ - 0 & 0 & 0 \\ - 0 & 0 & 0 - \end{bmatrix} - \end{align*} - \item Show that - \begin{align*} - \begin{bmatrix} - 0 & 1 & 0 \\ - 0 & 0 & 1 \\ - 0 & 0 & 0 - \end{bmatrix} - \sim - \begin{bmatrix} - 0 & 1 & 1 \\ - 0 & 0 & 1 \\ - 0 & 0 & 0 - \end{bmatrix} - \end{align*} - \item Find $n$ non-similar $n\times n$ nilpotent matrices for arbitrary $n$. - \item Show that - \begin{align*} - \begin{bmatrix} - 1 & 1 & 0 \\ - 0 & 1 & 1 \\ - 0 & 0 & 1 - \end{bmatrix} - \nsim - \begin{bmatrix} - 1 & 1 & 0 \\ - 0 & 1 & 0 \\ - 0 & 0 & 1 - \end{bmatrix} - \end{align*} - \end{enumerate} -\end{document} \ No newline at end of file diff --git a/research/04 Parigots exposition/problem 3.pdf b/research/04 Parigots exposition/problem 3.pdf deleted file mode 100644 index 1c5f34d1..00000000 Binary files a/research/04 Parigots exposition/problem 3.pdf and /dev/null differ diff --git a/research/04 Parigots exposition/problem 3.tex b/research/04 Parigots exposition/problem 3.tex deleted file mode 100644 index 86568626..00000000 --- a/research/04 Parigots exposition/problem 3.tex +++ /dev/null @@ -1,25 +0,0 @@ -\documentclass[12pt]{amsart} - -\usepackage{../AMC_style} -\usepackage{../Research} - -\DeclareMathOperator{\TT}{\boldface T} -\DeclareMathOperator{\A}{\boldface A} -\DeclareMathOperator{\B}{\boldface B} -\DeclareMathOperator{\PR}{P} - -\begin{document} - - Let $T$ be a linear transformation $T: V \arr V$, that is $W$-invariant for some subspace $W$ - \begin{enumerate} - \item Suppose $T$ is invertible. Show that $(\bar T)^{-1} = \overline {T^{-1}} $. - \item Show that $T^t$ is $W^0$-invariant (don't assume invertible anymore). - \end{enumerate} - - Let $S$ be a linear transformation $S: V \arr W$ - - \begin{enumerate} - \item Show that if $S$ is injective then $S^t$ is surjective. - \item Show that $\ker S^t = (\im S)^0$. - \end{enumerate} -\end{document} \ No newline at end of file diff --git a/research/04 Parigots exposition/problem 4.pdf b/research/04 Parigots exposition/problem 4.pdf deleted file mode 100644 index 98943948..00000000 Binary files a/research/04 Parigots exposition/problem 4.pdf and /dev/null differ diff --git a/research/04 Parigots exposition/problem 4.tex b/research/04 Parigots exposition/problem 4.tex deleted file mode 100644 index 5d819226..00000000 --- a/research/04 Parigots exposition/problem 4.tex +++ /dev/null @@ -1,25 +0,0 @@ -\documentclass[12pt]{amsart} - -\usepackage{../AMC_style} -\usepackage{../Research} - -\DeclareMathOperator{\TT}{\boldface T} -\DeclareMathOperator{\A}{\boldface A} -\DeclareMathOperator{\B}{\boldface B} -\DeclareMathOperator{\spn}{span} - -\begin{document} - - - \begin{enumerate} - \item Let $A \in M_n(F)$. Define $$V_A = \spn\{I, A, A^2, A^3, \ldots \} \subset M_n(F)$$ - Show that $\dim(V_A) = \deg(q_A)$ \\ - \item Now suppose that $A,B \in M_n(F)$ commute, that is $AB = BA$. Define - $$V_{A,B} = \spn\{A^i B^j \mid i,j \in \N\} \subset M_n(F)$$ - Show that $$\dim(V_{A,B}) \leq \dim(V_A) \dim(V_B)$$\\ - \item Using previous results show that - $$ \deg(q_{A+B}) \leq \deg(q_A) \deg(q_B) $$ - $$ \deg(q_{AB}) \leq \deg(q_A) \deg(q_B) $$ - \end{enumerate} - -\end{document} \ No newline at end of file diff --git a/research/04 Parigots exposition/problem.pdf b/research/04 Parigots exposition/problem.pdf deleted file mode 100644 index 1633e975..00000000 Binary files a/research/04 Parigots exposition/problem.pdf and /dev/null differ diff --git a/research/04 Parigots exposition/problem.tex b/research/04 Parigots exposition/problem.tex deleted file mode 100644 index 4ee7b304..00000000 --- a/research/04 Parigots exposition/problem.tex +++ /dev/null @@ -1,24 +0,0 @@ -\documentclass{amsart} - -\usepackage{../AMC_style} -\usepackage{../Research} - -\DeclareMathOperator{\TT}{\boldface T} -\DeclareMathOperator{\A}{\boldface A} -\DeclareMathOperator{\B}{\boldface B} -\DeclareMathOperator{\PR}{P} - -\begin{document} - Let $S = (s_1, s_2, \ldots, s_n)$ be a collection of vectors in a vector space $V$. - Suppose we also have $W \subseteq V$ a subspace. - Denote $\bar S = (\bar s_1, \bar s_2, \ldots, \bar s_n)$ where $\bar s_i = s_i + W$. - So $\bar S$ is a collection of vectors in $V/W$. - \begin{enumerate} - \item Show that if $S$ spans $V$ then $\bar S$ spans $V/W$. - \item Give an example of $S, V, W$ where $\bar S$ spans $V/W$ but $S$ doesn't span $V$. - \item Is it true that if $S$ is linearly independent in $V$ then $\bar S$ is linearly independent in $V/W$? - Prove it or give a counterexample. - \item Is it true that if $\bar S$ is linearly independent in $V/W$ then $S$ is linearly independent in $V$? - Prove it or give a counterexample. - \end{enumerate} -\end{document} \ No newline at end of file