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Abelian Group problems and solutions


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  • Diagonalizable Matrix with Eigenvalue 1, -1Diagonalizable Matrix with Eigenvalue 1, -1 Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues. Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix. (Stanford University Linear Algebra Exam) See below for a generalized problem. Hint. Diagonalize the […]
  • Determine a Value of Linear Transformation From $\R^3$ to $\R^2$Determine a Value of Linear Transformation From $\R^3$ to $\R^2$ Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that \[ T\left(\, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\,\right) =\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{ and }T\left(\, \begin{bmatrix} 0 \\ 1 \\ 1 […]
  • Express a Vector as a Linear Combination of Given Three VectorsExpress a Vector as a Linear Combination of Given Three Vectors Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 […]
  • Solving a System of Linear Equations Using Gaussian EliminationSolving a System of Linear Equations Using Gaussian Elimination Solve the following system of linear equations using Gaussian elimination. \begin{align*} x+2y+3z &=4 \\ 5x+6y+7z &=8\\ 9x+10y+11z &=12 \end{align*} Elementary row operations The three elementary row operations on a matrix are defined as […]
  • A Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as GeneratorA Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as Generator Let $R$ be a ring with $1$. A nonzero $R$-module $M$ is called irreducible if $0$ and $M$ are the only submodules of $M$. (It is also called a simple module.) (a) Prove that a nonzero $R$-module $M$ is irreducible if and only if $M$ is a cyclic module with any nonzero element […]
  • A Group Homomorphism and an Abelian GroupA Group Homomorphism and an Abelian Group Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$. Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ […]
  • Eigenvalues of a Hermitian Matrix are Real NumbersEigenvalues of a Hermitian Matrix are Real Numbers Show that eigenvalues of a Hermitian matrix $A$ are real numbers. (The Ohio State University Linear Algebra Exam Problem)   We give two proofs. These two proofs are essentially the same. The second proof is a bit simpler and concise compared to the first one. […]
  • The Quotient by the Kernel Induces an Injective HomomorphismThe Quotient by the Kernel Induces an Injective Homomorphism Let $G$ and $G'$ be a group and let $\phi:G \to G'$ be a group homomorphism.  Show that $\phi$ induces an injective homomorphism from $G/\ker{\phi} \to G'$.   Outline. Define $\tilde{\phi}([g])=\phi(g)$ and show that this is well-defined. Show […]

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