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…
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…
Homogeneous systems of linear equations in two unknowns with the same solutions are equivalent
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.6 Solution: Write the two systems as follows: $$ \begin{array}{c} a_{11}x+a_{12}y=0\\ a_{21}x+a_{22}y=0\\ \vdots\\ a_{m1}x+a_{m2}y=0 \end{array} \quad\quad \begin{array}{c} b_{11}x+b_{12}y=0\\ b_{21}x+b_{22}y=0\\ \vdots\\…
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…