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• If $R$ is a Noetherian Ring and $f:R\to R’$ is a Surjective Homomorphism, then $R’$ is Noetherian Suppose that $f:R\to R'$ is a surjective ring homomorphism. Prove that if $R$ is a Noetherian ring, then so is $R'$.   Definition. A ring $S$ is Noetherian if for every ascending chain of ideals of $S$ \[I_1 \subset I_2 \subset \cdots \subset I_k \subset […]
• Is the Set of Nilpotent Element an Ideal? Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$? If so, prove it. Otherwise give a counterexample.   Proof. We give a counterexample. Let $R$ be the noncommutative ring of $2\times 2$ matrices with real […]
• Find the Distance Between Two Vectors if the Lengths and the Dot Product are Given Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are \[\|\mathbf{a}\|=\|\mathbf{b}\|=1\] and the inner product \[\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.\] Then determine the length $\|\mathbf{a}-\mathbf{b}\|$. (Note […]
• Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations. (a) $\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]$. (b) […]
• Every Complex Matrix Can Be Written as $A=B+iC$, where $B, C$ are Hermitian Matrices (a) Prove that each complex $n\times n$ matrix $A$ can be written as \[A=B+iC,\] where $B$ and $C$ are Hermitian matrices. (b) Write the complex matrix \[A=\begin{bmatrix} i & 6\\ 2-i& 1+i \end{bmatrix}\] as a sum $A=B+iC$, where $B$ and $C$ are Hermitian […]
• The Set $ \{ a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$ is a Subspace in $C(\R)$ Let $C(\mathbb{R})$ be the vector space of real-valued functions on $\mathbb{R}$. Consider the set of functions $W = \{ f(x) = a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$. Prove that $W$ is a vector subspace of $C(\mathbb{R})$.   Proof. We verify […]
• The Order of a Conjugacy Class Divides the Order of the Group Let $G$ be a finite group. The centralizer of an element $a$ of $G$ is defined to be \[C_G(a)=\{g\in G \mid ga=ag\}.\] A conjugacy class is a set of the form \[\Cl(a)=\{bab^{-1} \mid b\in G\}\] for some $a\in G$. (a) Prove that the centralizer of an element of $a$ […]
• Find an Orthonormal Basis of $\R^3$ Containing a Given Vector Let $\mathbf{v}_1=\begin{bmatrix} 2/3 \\ 2/3 \\ 1/3 \end{bmatrix}$ be a vector in $\R^3$. Find an orthonormal basis for $\R^3$ containing the vector $\mathbf{v}_1$.   The first solution uses the Gram-Schumidt orthogonalization process. On the other hand, the second […]