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• Any Automorphism of the Field of Real Numbers Must be the Identity Map Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.   Proof. We prove the problem by proving the following sequence of claims. Let $\phi:\R \to \R$ be an automorphism of the field of real numbers […]
• Are Linear Transformations of Derivatives and Integrations Linearly Independent? Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation). Let $V$ be the vector space of all linear transformations from $W$ to $W$. The addition and the scalar multiplication of $V$ […]
• If the Nullity of a Linear Transformation is Zero, then Linearly Independent Vectors are Mapped to Linearly Independent Vectors Let $T: \R^n \to \R^m$ be a linear transformation. Suppose that the nullity of $T$ is zero. If $\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then show that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a […]
• True or False: If $A, B$ are 2 by 2 Matrices such that $(AB)^2=O$, then $(BA)^2=O$ Let $A$ and $B$ be $2\times 2$ matrices such that $(AB)^2=O$, where $O$ is the $2\times 2$ zero matrix. Determine whether $(BA)^2$ must be $O$ as well. If so, prove it. If not, give a counter example.   Proof. It is true that the matrix $(BA)^2$ must be the zero […]
• Probability of Getting Two Red Balls From the Chosen Box There are two boxes containing red and blue balls. Let us call the boxes Box A and Box B. Each box contains the same number of red and blue balls. More specifically, Box A has 5 red balls and 5 blue balls. Box B has 20 red balls and 20 blue balls. You choose one box. Then draw two […]
• Group 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 […]
• The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$. Then find the matrix representation of the linear transformation $T$ with respect to the […]
• If $ab=1$ in a Ring, then $ba=1$ when $a$ or $b$ is Not a Zero Divisor Let $R$ be a ring with $1\neq 0$. Let $a, b\in R$ such that $ab=1$. (a) Prove that if $a$ is not a zero divisor, then $ba=1$. (b) Prove that if $b$ is not a zero divisor, then $ba=1$.   Definition. An element $x\in R$ is called a zero divisor if there exists a […]