# UFD

by Yu ·

Add to solve later

Add to solve later

Add to solve later

### More from my site

- Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2 (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1. (b) Find all such matrices with rank 2. Solution. (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1. First we look at the rank 1 case. […]
- Quiz 10. Find Orthogonal Basis / Find Value of Linear Transformation (a) Let $S=\{\mathbf{v}_1, \mathbf{v}_2\}$ be the set of the following vectors in $\R^4$. \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}.\] […]
- Find All the Eigenvalues of Power of Matrix and Inverse Matrix Let \[A=\begin{bmatrix} 3 & -12 & 4 \\ -1 &0 &-2 \\ -1 & 5 & -1 \end{bmatrix}.\] Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$. Proof. We first determine all the eigenvalues of the matrix […]
- Basis with Respect to Which the Matrix for Linear Transformation is Diagonal Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by \[T(ax+b)=(3a+b)x+a+3,\] for any $ax+b\in P_1$. (a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation […]
- Rotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.\] (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
- The Powers of the Matrix with Cosine and Sine Functions Prove the following identity for any positive integer $n$. \[\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos […]
- Prove that any Algebraic Closed Field is Infinite Prove that any algebraic closed field is infinite. Definition. A field $F$ is said to be algebraically closed if each non-constant polynomial in $F[x]$ has a root in $F$. Proof. Let $F$ be a finite field and consider the polynomial \[f(x)=1+\prod_{a\in […]
- Show that Two Fields are Equal: $\Q(\sqrt{2}, \sqrt{3})= \Q(\sqrt{2}+\sqrt{3})$ Show that fields $\Q(\sqrt{2}+\sqrt{3})$ and $\Q(\sqrt{2}, \sqrt{3})$ are equal. Proof. It follows from $\sqrt{2}+\sqrt{3} \in \Q(\sqrt{2}, \sqrt{3})$ that we have $\Q(\sqrt{2}+\sqrt{3})\subset \Q(\sqrt{2}, \sqrt{3})$. To show the reverse inclusion, […]