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Diagonalization Problems and Solutions in Linear Algebra


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  • Matrix Operations with TransposeMatrix Operations with Transpose Calculate the following expressions, using the following matrices: \[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}\] (a) $A B^\trans + \mathbf{v} […]
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  • Galois Group of the Polynomial $x^2-2$Galois Group of the Polynomial $x^2-2$ Let $\Q$ be the field of rational numbers. (a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$? (b) Find the Galois group of $f(x)$ over $\Q$.   Solution. (a) The polynomial $f(x)=x^2-2$ is separable over $\Q$ The roots of the polynomial $f(x)$ are $\pm […]
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  • Equivalent Conditions For a Prime Ideal in a Commutative RingEquivalent Conditions For a Prime Ideal in a Commutative Ring Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent: (a) The ideal $P$ is a prime ideal. (b) For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.   Proof. […]
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  • Group of Order $pq$ is Either Abelian or the Center is TrivialGroup of Order $pq$ is Either Abelian or the Center is Trivial Let $G$ be a group of order $|G|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers. Then show that $G$ is either abelian group or the center $Z(G)=1$. Hint. Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […]
  • Is a Set of All Nilpotent Matrix a Vector Space?Is a Set of All Nilpotent Matrix a Vector Space? Let $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer. Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.   Definition. An matrix $A$ is a nilpotent matrix if […]

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