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Abelian Group problems and solutions


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  • The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$.The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$. Let \[R=\left\{\, \begin{bmatrix} a & b\\ 0& a \end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\] Then the usual matrix addition and multiplication make $R$ an ring. Let \[J=\left\{\, \begin{bmatrix} 0 & b\\ 0& 0 \end{bmatrix} […]
  • A Group Homomorphism is Injective if and only if MonicA Group Homomorphism is Injective if and only if Monic Let $f:G\to G'$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$. Then prove that a group homomorphism $f: G \to G'$ is injective if and only if it is […]
  • If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent?If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent? Consider the following system of linear equations: \begin{align*} ax_1+bx_2 &=c\\ dx_1+ex_2 &=f\\ gx_1+hx_2 &=i. \end{align*} (a) Write down the augmented matrix. (b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? […]
  • Compute the Determinant of a Magic SquareCompute the Determinant of a Magic Square Let \[ A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} . \] Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic […]
  • A Condition that a Commutator Group is a Normal SubgroupA 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 […]
  • Diagonalize the Upper Triangular Matrix and Find the Power of the MatrixDiagonalize the Upper Triangular Matrix and Find the Power of the Matrix Consider the $2\times 2$ complex matrix \[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$. (d) Using the result of the […]
  • Find a Polynomial Satisfying the Given Conditions on DerivativesFind a Polynomial Satisfying the Given Conditions on Derivatives Find a cubic polynomial \[p(x)=a+bx+cx^2+dx^3\] such that $p(1)=1, p'(1)=5, p(-1)=3$, and $ p'(-1)=1$.   Solution. By differentiating $p(x)$, we obtain \[p'(x)=b+2cx+3dx^2.\] Thus the given conditions are […]
  • Any Vector is a Linear Combination of Basis Vectors UniquelyAny Vector is a Linear Combination of Basis Vectors Uniquely Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as \[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\] where $c_1, c_2, c_3$ are […]

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