# field-theory-2

• Subgroup of Finite Index Contains a Normal Subgroup of Finite Index Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.   Proof. The group $G$ acts on the set of left cosets $G/H$ by left multiplication. Hence […]
• All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$ Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.   Solution. Let $T:\R^2 \to \R^2$ be a linear transformation that maps the line $y=x$ to the line $y=-x$. Note that the linear […]
• Companion Matrix for a Polynomial Consider a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are real numbers. Define the matrix A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & […] • Prove Vector Space Properties Using Vector Space Axioms Using the axiom of a vector space, prove the following properties. Let V be a vector space over \R. Let u, v, w\in V. (a) If u+v=u+w, then v=w. (b) If v+u=w+u, then v=w. (c) The zero vector \mathbf{0} is unique. (d) For each v\in V, the additive inverse […] • Every 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 […] • Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible? A square matrix A is called nilpotent if some power of A is the zero matrix. Namely, A is nilpotent if there exists a positive integer k such that A^k=O, where O is the zero matrix. Suppose that A is a nilpotent matrix and let B be an invertible matrix of […] • Expectation, Variance, and Standard Deviation of Bernoulli Random Variables A random variable X is said to be a Bernoulli random variable if its probability mass function is given by \begin{align*} P(X=0) &= 1-p\\ P(X=1) & = p \end{align*} for some real number 0 \leq p \leq 1. (1) Find the expectation of the Bernoulli random variable X […] • Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let \calP_3 be the vector space of all polynomials of degree 3 or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}, where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (a) […]