# Recent questions in Forms of linear equations

Forms of linear equations

### Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$ $$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$ $$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$

Forms of linear equations

### Write the system of linear equations in the form $$\displaystyle{A}{x}={b}$$ and solve this matrix equation for x. $$\displaystyle-{x}_{{{1}}}+{x}_{{{2}}}={4}$$ $$\displaystyle-{2}{x}_{{{1}}}+{x}_{{{2}}}={0}$$

Forms of linear equations

### Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations $$\begin{cases}9x-3y+z=13 \\ 12x-8z=5 \\ 3x+4y-z =6 \end{cases}$$

Forms of linear equations

### Let B be a $$(4\times3)(4\times3)$$ matrix in reduced echelon form. a) If B has three nonzero rows, then determine the form of B. b) Suppose that a system of 4 linear equations in 2 unknowns has augmented matrix A, where A is a $$(4\times3)(4\times3)$$ matrix row equivalent to B. Demonstrate that the system of equations is inconsistent.

Forms of linear equations

### Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form $$x=at+b , y=ct+d , z=et+f$$ $$2x_1+x_2+x_3=3$$ $$-2x_1+x_2-x_3=1$$

Forms of linear equations

### The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $$\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}$$

Forms of linear equations

### Determine whether the given $$\displaystyle{\left({2}\ \times\ {3}\right)}$$ system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form $$\displaystyle{x}={a}{t}\ +\ {b},\ {y}={c}{t}\ +\ {d},\ {z}={e}{t}\ +\ {f}.$$ $$\displaystyle{x}_{{{1}}}\ +\ {2}{x}_{{{2}}}\ -\ {x}_{{{3}}}={2}$$ $$\displaystyle{x}_{{{1}}}\ +\ {x}_{{{2}}}\ +\ {x}_{{{3}}}={3}$$

Forms of linear equations

### The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$ has the given eigenvalues and eigenspace bases. Find the general solution for the system $$\lambda_1=2i \Rightarrow \left\{ \begin{bmatrix}1+i\\ 2-i \end{bmatrix} \right\} , \lambda_2=-2i \Rightarrow \left\{ \begin{bmatrix}1-i\\ 2+i \end{bmatrix} \right\}$$

Forms of linear equations

### The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$  has the given eigenvalues and eigenspace bases. Find the general solution for the system  $$\lambda1=3\Rightarrow \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$$ $$\lambda2=0\Rightarrow \begin{bmatrix} 1 \\ 5 \\ 1 \end{bmatrix}\begin{bmatrix}2 \\ 1 \\ 4 \end{bmatrix}$$

Forms of linear equations

### Write the vector form of the general solution of the given system of linear equations. $$x_1+2x_2-x_3=0$$ $$x_1+x_2+x_3=0$$ $$x_1+3x_2-3x_3=0$$

Forms of linear equations

### The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$$

Forms of linear equations