# 2568syllabus_f16

2568syllabus_f16

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• Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where \begin{align*} p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\ p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2. \end{align*} (a) Find […]
• Row Equivalence of Matrices is Transitive If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$? If so, then prove it. If not, then provide a counterexample.   Definition (Row […]
• Finite Group and a Unique Solution of an Equation Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that $b^m=a.$   We give two proofs. Proof 1. Since $m$ and $n$ are relatively prime […]
• Properties of Nonsingular and Singular Matrices An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$. Otherwise $A$ is called singular. (a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is […]
• Every Integral Domain Artinian Ring is a Field Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring. Prove that $R$ is a field.   Definition (Artinian ring). A ring $R$ is called Artinian if it satisfies the defending chain condition on ideals. That is, whenever we have […]
• Galois Group of the Polynomial $x^2-2$ Let $\Q$ be the field of rational numbers. (a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$? (b) Find the Galois group of $f(x)$ over $\Q$.   Solution. (a) The polynomial $f(x)=x^2-2$ is separable over $\Q$ The roots of the polynomial $f(x)$ are $\pm […] • Equivalent Conditions For a Prime Ideal in a Commutative Ring Let$R$be a commutative ring and let$P$be an ideal of$R$. Prove that the following statements are equivalent: (a) The ideal$P$is a prime ideal. (b) For any two ideals$I$and$J$, if$IJ \subset P$then we have either$I \subset P$or$J \subset P$. Proof. […] • A Condition that a Commutator Group is a Normal Subgroup Let$H$be a normal subgroup of a group$G$. Then show that$N:=[H, G]$is a subgroup of$H$and$N \triangleleft G$. Here$[H, G]$is a subgroup of$G$generated by commutators$[h,k]:=hkh^{-1}k^{-1}$. In particular, the commutator subgroup$[G, G]\$ is a normal subgroup of […]