linear-algebra-eyecatch

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Linear Algebra Problems and Solutions


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  • Exponential Functions Form a Basis of a Vector SpaceExponential 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 MatrixA 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 MatrixThe 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 AbelianA 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 ZeroA 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 8All 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 DomainThe 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 RingPrime 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 […]

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