## Verify the set with addition and multiplication is a field

Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.5 Solution: We must check the nine conditions on pages 1-2: An operation is commutative if the table is symmetric…

Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.5 Solution: We must check the nine conditions on pages 1-2: An operation is commutative if the table is symmetric…

Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.4 Solution: These systems are not equivalent. Call the two equations in the first system $E_1$ and $E_2$ and the…

Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.3 Solution: Yes the two systems are equivalent. We show this by writing each equation of the first system in…

Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.2 Solution: Yes the two systems are equivalent. We show this by writing each equation of the first system in…

Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.1 Solution: Let $F=\{x+y\sqrt{2}\mid x,y\in\mb Q\}$. Then we must show six things: $0$ is in $F$ $1$ is in $F$…

Vandermonde matrices have the following form:$$A_n=\begin{bmatrix} 1&x_1&x_1^2&\cdots&x_1^{n-1}\\ 1&x_2&x_2^2&\cdots&x_2^{n-1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&x_n&x_n^2&\cdots&x_n^{n-1} \end{bmatrix},$$ where $A_n=[a_{ij}]_{i,j=1}^n$ is an $n\times n$ matrices and the element $a_{ij}=x_i^{j-1}$. The transpose $A_n^T$ is also called a Vandermonde…

Let $A$ be an $n\times n$ matrix. If there exists a positive integer $q$ such that \begin{equation}\label{eq:1}A^{q}=0,\end{equation} then we call $A$ a nilpotent matrix, meaning that one of its powers…

Chapter 1. Linear Equations 1.1 Fields (no exercises) 1.2 Systems of Linear Equations (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) 1.3 Matrices and Elementary Row Operations (#1) (#2) (#3)…