Solve general linear equations with $2\times 2$ matrix
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.8 Solution: (a) In this case the system of equations is \begin{alignat*}{1} 0\cdot x_1 + 0\cdot x_2 &= 0\\ 0\cdot…
Interchange of two rows can be accomplished by elementary row operations of other types
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.7 Solution: Write the matrix as $$\left[\begin{array}{c} R_1\\ R_2\\ R_3\\ \vdots\\ R_n \end{array}\right].$$WOLOG we'll show how to exchange rows $R_1$…
$2\times 2$ Row-reduced Matrix
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.6 Solution: Case $a\not=0$: Then to be in row-reduced form it must be that $a=1$ and $A=\left[\begin{array}{cc}1&b\\c&d\end{array}\right]$ which implies $c=0$,…
Prove that the following two matrices are not row-equivalent
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.5 Solution: Call the first matrix $A$ and the second matrix $B$. The matrix $A$ is row-equivalent to $$A'=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]$$ and…
Find a row-reduced matrix to the given one
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.4 Solution: We have $$A\rightarrow\left[\begin{array}{ccc} 1 & -2 & 1\\ i & -(1+i) & 0\\ 1 & 2i & -1…
Find all solutions to the systems of equations by row-reducing (3)
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.3 Solution: The system $AX=2X$ is $$\left[\begin{array}{ccc}6&-4&0\\4&-2&0\\-1&0&3\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]=2\left[\begin{array}{c}x\\y\\z\end{array}\right]$$which is the same as \begin{alignat*}{1} 6x-4y&=2x\\ 4x-2y&=2y\\ -x+3z&=2z \end{alignat*}which is equivalent to \begin{alignat*}{1}…
Find all solutions to the systems of equations by row-reducing (2)
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.2 Solution: We have$$\rightarrow\left[\begin{array}{ccc}1&-3&0\\2&1&1\\3&-1&2\end{array}\right]\rightarrow\left[\begin{array}{ccc}1&-3&0\\0&7&1\\0&8&2\end{array}\right]\rightarrow\left[\begin{array}{ccc}1&-3&0\\0&1&1/7\\0&8&2\end{array}\right]$$ $$\rightarrow\left[\begin{array}{ccc}1&0&3/7\\0&1&1/7\\0&0&6/7\end{array}\right]\rightarrow\left[\begin{array}{ccc}1&0&3/7\\0&1&1/7\\0&0&1\end{array}\right]\rightarrow\left[\begin{array}{ccc}1&0&0\\0&1&1/7\\0&0&1\end{array}\right].$$Thus $A$ is row-equivalent to the identity matrix. It follows that the only solution to the…
Find all solutions to the systems of equations by row-reducing (1)
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.1 Solution: The matrix of coefficients is $$\left[\begin{array}{cc}1-i&-i\\2&1-i\end{array}\right].$$Row reducing $$\rightarrow \left[\begin{array}{cc}2&1-i\\1-i&-i\end{array}\right]\rightarrow\left[\begin{array}{cc}2&1-i\\0&0\end{array}\right] $$Thus $2x_1+(1-i)x_2=0$. Thus for any $x_2\in\mathbb C$, $(\frac12(i-1)x_2,x_2)$ is…
Each field of characteristic zero contains a copy of the rational number field
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.8 Solution: Call the additive and multiplicative identities of $\mb F$ $0_{\mb F}$ and $1_{\mb F}$ respectively. Define $n_{\mb F}$…
Each subfield of the field of complex numbers contains every rational number
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.7 Solution: Every subfield of $\C$ has characterisitc zero since if $\mb F$ is a subfield then $1\in \mb F$…