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…
Are the following two systems of linear equations equivalent (3)
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…
Are the following two systems of linear equations equivalent (2)
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…
Are the following two systems of linear equations equivalent (1)
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…
Verify that the set of complex numbers is a subfield of C
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 $F$…
Special Matrix (8) Vandermonde Matrix
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…
Special Matrix (1) Nilpotent Matrix
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…
Solution to Linear Algebra Hoffman & Kunze Second Edition
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)…