# pi2018

by Yu ·

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### More from my site

- The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$. Then prove that $V$ is a subspace of $\R^n$. Proof. To prove that $V=\{\mathbf{0}\}$ is a subspace of $\R^n$, we check the following subspace […]
- How Many Solutions for $x+x=1$ in a Ring? Is there a (not necessarily commutative) ring $R$ with $1$ such that the equation \[x+x=1 \] has more than one solutions $x\in R$? Solution. We claim that there is at most one solution $x$ in the ring $R$. Suppose that we have two solutions $r, s \in R$. That is, we […]
- Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup Let $G$ be a group and $H$ and $K$ be subgroups of $G$. For $h \in H$, and $k \in K$, we define the commutator $[h, k]:=hkh^{-1}k^{-1}$. Let $[H,K]$ be a subgroup of $G$ generated by all such commutators. Show that if $H$ and $K$ are normal subgroups of $G$, then the subgroup […]
- Conjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the Set Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$. For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$. Proof. $(\subset)$ We first show that $gC_G(X)g^{-1} \subset C_G(gXg^{-1})$. Take any $h\in C_G(X)$. Then for […]
- Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism? Let $V$ denote the vector space of all real $2\times 2$ matrices. Suppose that the linear transformation from $V$ to $V$ is given as below. \[T(A)=\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}A-A\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}.\] Prove or […]
- Find the Inverse Matrix of a $3\times 3$ Matrix if Exists Find the inverse matrix of \[A=\begin{bmatrix} 1 & 1 & 2 \\ 0 &0 &1 \\ 1 & 0 & 1 \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason. (The Ohio State University, Linear Algebra Midterm Exam […]
- Similar Matrices Have the Same Eigenvalues Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same. Proof. We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]
- Basis with Respect to Which the Matrix for Linear Transformation is Diagonal Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by \[T(ax+b)=(3a+b)x+a+3,\] for any $ax+b\in P_1$. (a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation […]