# Group-Theory2

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- Given All Eigenvalues and Eigenspaces, Compute a Matrix Product Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces \[E_2=\Span\left \{\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix} 1 \\ 2 \\ 1 \\ 1 […]
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- A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$ An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that $AB=I$, and $BA=I$, where $I$ is the $n\times n$ identity matrix. If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted […]
- A 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 […]
- Differentiation is a Linear Transformation Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients. (a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by \[T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)\] for any $f(x)\in […]
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