# linear-algebra-eyecatch

### More from my site

• Exponential Functions Form a Basis of a Vector Space Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let $V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}$ be a subset in $C[-1, 1]$. (a) Prove that $V$ is a subspace of $C[-1, 1]$. (b) […]
• A Recursive Relationship for a Power of a Matrix Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation $A^{n} = b_n A + c_n I ,$ where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]
• The Inverse Matrix of the Transpose is the Transpose of the Inverse Matrix Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$. Namely, show […]
• A Group of Order the Square of a Prime is Abelian Suppose the order of a group $G$ is $p^2$, where $p$ is a prime number. Show that (a) the group $G$ is an abelian group, and (b) the group $G$ is isomorphic to either $\Zmod{p^2}$ or $\Zmod{p} \times \Zmod{p}$ without using the fundamental theorem of abelian […]
• 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 $b = […] • All the Conjugacy Classes of the Dihedral Group$D_8$of Order 8 Determine all the conjugacy classes of the dihedral group $D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle$ of order$8$. Hint. You may directly compute the conjugates of each element but we are going to use the following theorem to simplify the […] • The Ideal$(x)$is Prime in the Polynomial Ring$R[x]$if and only if the Ring$R$is an Integral Domain Let$R$be a commutative ring with$1$. Prove that the principal ideal$(x)$generated by the element$x$in the polynomial ring$R[x]$is a prime ideal if and only if$R$is an integral domain. Prove also that the ideal$(x)$is a maximal ideal if and only if$R$is a […] • Prime Ideal is Irreducible in a Commutative Ring Let$R$be a commutative ring. An ideal$I$of$R$is said to be irreducible if it cannot be written as an intersection of two ideals of$R$which are strictly larger than$I$. Prove that if$\frakp$is a prime ideal of the commutative ring$R$, then$\frakp\$ is […]