field-theory-eyecatch

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Field theory eye catch


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  • In which $\R^k$, are the Nullspace and Range Subspaces?In which $\R^k$, are the Nullspace and Range Subspaces? Let $A$ be an $m \times n$ matrix. Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.   Solution. For an $m \times n$ matrix $A$, the […]
  • If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity MatrixIf a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix. Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.   Proof. Because $A$ has rank $n$, we know that the $n \times n$ […]
  • Finite Group and a Unique Solution of an EquationFinite 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 […]
  • A Group of Linear FunctionsA Group of Linear Functions Define the functions $f_{a,b}(x)=ax+b$, where $a, b \in \R$ and $a>0$. Show that $G:=\{ f_{a,b} \mid a, b \in \R, a>0\}$ is a group . The group operation is function composition. Steps. Check one by one the followings. The group operation on $G$ is […]
  • The Number of Elements Satisfying $g^5=e$ in a Finite Group is OddThe Number of Elements Satisfying $g^5=e$ in a Finite Group is Odd Let $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$. Prove that the number of elements in $S$ is odd.   Proof. Let $g\neq e$ be an element in the group $G$ such that $g^5=e$. As […]
  • Inverse Map of a Bijective Homomorphism is a Group HomomorphismInverse Map of a Bijective Homomorphism is a Group Homomorphism Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism. Suppose that $f:G\to H$ is bijective. Then there exists a map $\psi:H\to G$ such that \[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\] Then prove that $\psi:H \to G$ is also a group […]
  • Polynomial $(x-1)(x-2)\cdots (x-n)-1$ is Irreducible Over the Ring of Integers $\Z$Polynomial $(x-1)(x-2)\cdots (x-n)-1$ is Irreducible Over the Ring of Integers $\Z$ For each positive integer $n$, prove that the polynomial \[(x-1)(x-2)\cdots (x-n)-1\] is irreducible over the ring of integers $\Z$.   Proof. Note that the given polynomial has degree $n$. Suppose that the polynomial is reducible over $\Z$ and it decomposes as […]
  • 12 Examples of Subsets that Are Not Subspaces of Vector Spaces12 Examples of Subsets that Are Not Subspaces of Vector Spaces Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace. (1) \[S_1=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\] in […]

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