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## 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…