# abelian-groups-eye-catch

• Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$ (a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where $B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}.$ (b) Find the inverse matrix of $A$.   Solution (a) Find a $3\times 3$ nonsingular matrix $A$. Assume […]
• A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space (a) Suppose that $A$ is an $n\times n$ real symmetric positive definite matrix. Prove that $\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}$ defines an inner product on the vector space $\R^n$. (b) Let $A$ be an $n\times n$ real matrix. Suppose […]
• Inner Products, Lengths, and Distances of 3-Dimensional Real Vectors For this problem, use the real vectors $\mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} .$ Suppose that $\mathbf{v}_4$ is another vector which is […]
• Find the Matrix Representation of $T(f)(x) = f(x^2)$ if it is a Linear Transformation For an integer $n > 0$, let $\mathrm{P}_n$ denote the vector space of polynomials with real coefficients of degree $2$ or less. Define the map $T : \mathrm{P}_2 \rightarrow \mathrm{P}_4$ by $T(f)(x) = f(x^2).$ Determine if $T$ is a linear transformation. If it is, find […]
• Find a Formula for a Linear Transformation If $L:\R^2 \to \R^3$ is a linear transformation such that \begin{align*} L\left( \begin{bmatrix} 1 \\ 0 \end{bmatrix}\right) =\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \,\,\,\, L\left( \begin{bmatrix} 1 \\ 1 \end{bmatrix}\right) =\begin{bmatrix} 2 \\ 3 […]
• Find a Nonsingular Matrix Satisfying Some Relation Determine whether there exists a nonsingular matrix $A$ if $A^2=AB+2A,$ where $B$ is the following matrix. If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$. (a) $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 1 & 2 & […] • Primary Ideals, Prime Ideals, and Radical Ideals Let R be a commutative ring with unity. A proper ideal I of R is called primary if whenever ab \in I for a, b\in R, then either a\in I or b^n\in I for some positive integer n. (a) Prove that a prime ideal P of R is primary. (b) If P is a prime ideal and […] • Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less Let P_2 be the vector space over \R of all polynomials of degree 2 or less. Let S=\{p_1(x), p_2(x), p_3(x)\}, where \[p_1(x)=x^2+1, \quad p_2(x)=6x^2+x+2, \quad p_3(x)=3x^2+x.$ (a) Use the basis $B=\{x^2, x, 1\}$ of $P_2$ to prove that the set $S$ is a basis for […]