# abelian-group-eye-catch

• 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 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? 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 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 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 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 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 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 […]