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• Vector Space of Polynomials and a Basis of Its Subspace Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=1, &p_2(x)=x^2+x+1, \\ &p_3(x)=2x^2, &p_4(x)=x^2-x+1. \end{align*} (a) Use the basis $B=\{1, x, […]
• Is an Eigenvector of a Matrix an Eigenvector of its Inverse? Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$. (a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not. (b) Is $3\mathbf{v}$ an […]
• Determine Whether Each Set is a Basis for $\R^3$ Determine whether each of the following sets is a basis for $\R^3$. (a) $S=\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} […]
• Polynomial $x^p-x+a$ is Irreducible and Separable Over a Finite Field Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements. For any nonzero element $a\in \F_p$, prove that the polynomial \[f(x)=x^p-x+a\] is irreducible and separable over $F_p$. (Dummit and Foote "Abstract Algebra" Section 13.5 Exercise #5 on […]
• Transpose of a Matrix and Eigenvalues and Related Questions Let $A$ be an $n \times n$ real matrix. Prove the followings. (a) The matrix $AA^{\trans}$ is a symmetric matrix. (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal. (c) The matrix $AA^{\trans}$ is non-negative definite. (An $n\times n$ […]
• $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain. Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.   Proof. Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$. Define the function $\Psi:R[x,y] \to R[t]$ sending […]
• How to Find a Formula of the Power of a Matrix Let $A= \begin{bmatrix} 1 & 2\\ 2& 1 \end{bmatrix}$. Compute $A^n$ for any $n \in \N$. Plan. We diagonalize the matrix $A$ and use this Problem. Steps. Find eigenvalues and eigenvectors of the matrix $A$. Diagonalize the matrix $A$. Use […]
• Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$ Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.   Definition (Symmetric Matrix). A matrix $A$ is called symmetric if $A^{\trans}=A$. In terms of entries, an $n\times n$ matrix $A=(a_{ij})$ is […]