# vector-space

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• Non-Prime Ideal of Continuous Functions Let $R$ be the ring of all continuous functions on the interval $[0,1]$. Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$. Show that the set $I$ is an ideal of $R$ but is not a prime ideal.   Proof. We first show that $I$ is an ideal of […]
• Galois Group of the Polynomial $x^2-2$ Let $\Q$ be the field of rational numbers. (a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$? (b) Find the Galois group of $f(x)$ over $\Q$.   Solution. (a) The polynomial $f(x)=x^2-2$ is separable over $\Q$ The roots of the polynomial $f(x)$ are $\pm […] • Group of Order$pq$Has a Normal Sylow Subgroup and Solvable Let$p, q$be prime numbers such that$p>q$. If a group$G$has order$pq$, then show the followings. (a) The group$G$has a normal Sylow$p$-subgroup. (b) The group$G$is solvable. Definition/Hint For (a), apply Sylow's theorem. To review Sylow's theorem, […] • Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors Let$V$be a subset of$\R^4$consisting of vectors that are perpendicular to vectors$\mathbf{a}, \mathbf{b}$and$\mathbf{c}$, where $\mathbf{a}=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix} 1 \\ 1 […] • 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 […] • Order of the Product of Two Elements in an Abelian Group Let$G$be an abelian group with the identity element$1$. Let$a, b$be elements of$G$with order$m$and$n$, respectively. If$m$and$n$are relatively prime, then show that the order of the element$ab$is$mn$. Proof. Let$r$be the order of the element […] • Determine Trigonometric Functions with Given Conditions (a) Find a function $g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)$ such that$g(0) = g(\pi/2) = g(\pi) = 0$, where$a, b, c$are constants. (b) Find real numbers$a, b, c$such that the function $g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […] • Find the Distance Between Two Vectors if the Lengths and the Dot Product are Given Let \mathbf{a} and \mathbf{b} be vectors in \R^n such that their length are \[\|\mathbf{a}\|=\|\mathbf{b}\|=1$ and the inner product $\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.$ Then determine the length$\|\mathbf{a}-\mathbf{b}\|\$. (Note […]