Author: Linearity

Exercise 1.6.1 As in Exercise 4, Section 1.5, we row reduce and keep track of the elementary matrices involved. It takes nine steps to put $A$ in row-reduced form resulting…

Exercise 1.5.1 We have $$AB=\left[\begin{array}{c} 4\\4\end{array}\right],$$so $$ABC= \left[\begin{array}{c} 4\\4\end{array}\right]\cdot [1\ \ -1] =\left[\begin{array}{cc} 4 & -4\\ 4 & -4 \end{array}\right].$$and $$CBA=[1\ \ -1] \cdot \left[\begin{array}{c} 4\\4\end{array}\right] = [0].$$ Exercise 1.5.2…

Exercise 1.4.1 The coefficient matrix is $$\left[\begin{array}{ccc} \frac13 & 2 & -6\\ -4& 0& 5\\ -3&6&-13\\ -\frac73&2&-\frac83 \end{array}\right] $$This reduces as follows: $$\rightarrow\left[\begin{array}{ccc} 1 & 6 & -18\\ -4& 0&…

Exercise 1.3.1 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 a solution and these are all solutions. Exercise 1.3.2 We…

Exercise 1.2.1: Verify that the set of complex numbers described in Example 4 is a subfield of $\mathbb C$. Solution: Let $F=\{x+y\sqrt{2}\mid x,y\in\mb Q\}$. Then we must show six things:…