# triangle2018

by Yu · Published · Updated

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- If a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero Matrix Let $A$ be an $n\times n$ singular matrix. Then prove that there exists a nonzero $n\times n$ matrix $B$ such that \[AB=O,\] where $O$ is the $n\times n$ zero matrix. Definition. Recall that an $n \times n$ matrix $A$ is called singular if the […]
- Subspaces of the Vector Space of All Real Valued Function on the Interval Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not. (a) $S=\{f(x) \in V \mid f(0)=f(1)\}$. (b) $T=\{f(x) \in V \mid […]
- If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix Let $A, B$ be complex $2\times 2$ matrices satisfying the relation \[A=AB-BA.\] Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix. Hint. Find the trace of $A$. Use the Cayley-Hamilton theorem Proof. We first calculate the […]
- Use Lagrange’s Theorem to Prove Fermat’s Little Theorem Use Lagrange's Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat's Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$. Before the proof, let us recall Lagrange's Theorem. Lagrange's Theorem If $G$ is a […]
- Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$ Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that \[T(\mathbf{v}_1)=\begin{bmatrix} 1 \\ -2 \end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix} 3 \\ 1 […]
- Is the Given Subset of The Ring of Integer Matrices an Ideal? Let $R$ be the ring of all $2\times 2$ matrices with integer coefficients: \[R=\left\{\, \begin{bmatrix} a & b\\ c& d \end{bmatrix} \quad \middle| \quad a, b, c, d\in \Z \,\right\}.\] Let $S$ be the subset of $R$ given by \[S=\left\{\, \begin{bmatrix} s & […]
- $p$-Group Acting on a Finite Set and the Number of Fixed Points Let $P$ be a $p$-group acting on a finite set $X$. Let \[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \] The prove that \[|X^P|\equiv |X| \pmod{p}.\] Proof. Let $\calO(x)$ denote the orbit of $x\in X$ under the action of the group $P$. Let […]
- Find Inverse Matrices Using Adjoint Matrices Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix […]