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• Upper Bound of the Variance When a Random Variable is Bounded Let $c$ be a fixed positive number. Let $X$ be a random variable that takes values only between $0$ and $c$. This implies the probability $P(0 \leq X \leq c) = 1$. Then prove the next inequality about the variance $V(X)$. \[V(X) \leq \frac{c^2}{4}.\] Proof. Recall that […]
• If matrix product $AB$ is a square, then is $BA$ a square matrix? Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix. Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a […]
• The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$ Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by \[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\] That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$. Prove that the […]
• If a Matrix is the Product of Two Matrices, is it Invertible? (a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as \[A=BC,\] where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix. Prove that the matrix $A$ cannot be invertible. (b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be […]
• Ring Homomorphisms and Radical Ideals Let $R$ and $R'$ be commutative rings and let $f:R\to R'$ be a ring homomorphism. Let $I$ and $I'$ be ideals of $R$ and $R'$, respectively. (a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that $\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$ (c) Suppose that $f$ is […]
• If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$. Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix. Steps. Use the Jordan canonical form of the matrix $A$. We want […]
• Normal Nilpotent Matrix is Zero Matrix A complex square ($n\times n$) matrix $A$ is called normal if \[A^* A=A A^*,\] where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$. A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero […]
• A Recursive Relationship for a Power of a Matrix Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation \[ A^{n} = b_n A + c_n I , \] where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]