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  • Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions?Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions? Let $C (\mathbb{R})$ be the vector space of real functions. Define the map $T$ by $T(f)(x) = (f(x))^2$ for $f \in C(\mathbb{R})$. Determine if $T$ is a linear transformation or not. If it is, determine the range of $T$.   Solution. We claim that $T$ is not a […]
  • Find an Orthonormal Basis of the Range of a Linear TransformationFind an Orthonormal Basis of the Range of a Linear Transformation Let $T:\R^2 \to \R^3$ be a linear transformation given by \[T\left(\, \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \,\right) = \begin{bmatrix} x_1-x_2 \\ x_2 \\ x_1+ x_2 \end{bmatrix}.\] Find an orthonormal basis of the range of $T$. (The Ohio […]
  • Two Quadratic Fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are Not IsomorphicTwo Quadratic Fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are Not Isomorphic Prove that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.   Hint. Note that any homomorphism between fields over $\Q$ fixes $\Q$ pointwise. Proof. Assume that there is an isomorphism $\phi:\Q(\sqrt{2}) \to \Q(\sqrt{3})$. Let […]
  • Multiplicative Groups of Real Numbers and Complex Numbers are not IsomorphicMultiplicative Groups of Real Numbers and Complex Numbers are not Isomorphic Let $\R^{\times}=\R\setminus \{0\}$ be the multiplicative group of real numbers. Let $\C^{\times}=\C\setminus \{0\}$ be the multiplicative group of complex numbers. Then show that $\R^{\times}$ and $\C^{\times}$ are not isomorphic as groups.   Recall. Let $G$ and $K$ […]
  • Every Diagonalizable Nilpotent Matrix is the Zero MatrixEvery Diagonalizable Nilpotent Matrix is the Zero Matrix Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.   Definition (Nilpotent Matrix) A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$. Proof. Main Part Since $A$ is […]
  • For Fixed Matrices $R, S$, the Matrices $RAS$ form a SubspaceFor Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace Let $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$. Prove that $W$ is a vector subspace of $V$.   Proof. We verify the subspace criteria: the zero vector of $V$ is in $W$, and […]
  • Basic Exercise Problems in Module TheoryBasic Exercise Problems in Module Theory Let $R$ be a ring with $1$ and $M$ be a left $R$-module. (a) Prove that $0_Rm=0_M$ for all $m \in M$. Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$. To simplify the […]
  • A Homomorphism from the Additive Group of Integers to ItselfA Homomorphism from the Additive Group of Integers to Itself Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism. Then show that there exists an integer $a$ such that \[f(n)=an\] for any integer $n$.   Hint. Let us first recall the definition of a group homomorphism. A group homomorphism from a […]

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