# 2568syllabus_f16

• Determine When the Given Matrix Invertible For which choice(s) of the constant $k$ is the following matrix invertible? $A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &k \\ 1 & 4 & k^2 \end{bmatrix}.$   (Johns Hopkins University, Linear Algebra Exam)   Hint. An $n\times n$ matrix is […]
• A Line is a Subspace if and only if its $y$-Intercept is Zero Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by $\ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .$ Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = […] • Matrix of Linear Transformation with respect to a Basis Consisting of Eigenvectors Let$T$be the linear transformation from the vector space$\R^2$to$\R^2$itself given by $T\left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right)= \begin{bmatrix} 3x_1+x_2 \\ x_1+3x_2 \end{bmatrix}.$ (a) Verify that the […] • How Many Solutions for$x+x=1$in a Ring? Is there a (not necessarily commutative) ring$R$with$1$such that the equation $x+x=1$ has more than one solutions$x\in R$? Solution. We claim that there is at most one solution$x$in the ring$R$. Suppose that we have two solutions$r, s \in R$. That is, we […] • Automorphism Group of$\Q(\sqrt[3]{2})$Over$\Q$. Determine the automorphism group of$\Q(\sqrt[3]{2})$over$\Q$. Proof. Let$\sigma \in \Aut(\Q(\sqrt[3]{2}/\Q)$be an automorphism of$\Q(\sqrt[3]{2})$over$\Q$. Then$\sigma$is determined by the value$\sigma(\sqrt[3]{2})$since any element$\alpha$of$\Q(\sqrt[3]{2})$[…] • A Group Homomorphism and an Abelian Group Let$G$be a group. Define a map$f:G \to G$by sending each element$g \in G$to its inverse$g^{-1} \in G$. Show that$G$is an abelian group if and only if the map$f: G\to G$is a group homomorphism. Proof.$(\implies)$If$G$is an abelian group, then$f$[…] • Determine whether the Given 3 by 3 Matrices are Nonsingular Determine whether the following matrices are nonsingular or not. (a)$A=\begin{bmatrix} 1 & 0 & 1 \\ 2 &1 &2 \\ 1 & 0 & -1 \end{bmatrix}$. (b)$B=\begin{bmatrix} 2 & 1 & 2 \\ 1 &0 &1 \\ 4 & 1 & 4 \end{bmatrix}$. Solution. Recall that […] • Abelian Normal Subgroup, Intersection, and Product of Groups Let$G$be a group and let$A$be an abelian subgroup of$G$with$A \triangleleft G$. (That is,$A$is a normal subgroup of$G$.) If$B$is any subgroup of$G$, then show that $A \cap B \triangleleft AB.$ Proof. First of all, since$A \triangleleft G\$, the […]