Each Archetype (Archetypes) that is a system of equations has a corresponding homogeneous system with the same coefficient matrix. Compute the set of solutions for each. Notice that these solution sets are the null spaces of the coefficient matrices.
Archetype A,
Archetype B,
Archetype C,
Archetype D/Archetype E,
Archetype F,
Archetype G/Archetype H,
Archetype I,
Archetype J
Archetype K and Archetype L are simply $5\times 5$ matrices (i.e. they are not systems of equations). Compute the null space of each matrix.
For Exercises C21-C23, solve the given homogeneous linear system. Compare your results to the results of the corresponding exercise in Section TSS.
\begin{align*} x_1 + 4x_2 + 3x_3 - x_4 &= 0\\ x_1 - x_2 + x_3 + 2x_4 &= 0\\ 4x_1 + x_2 + 6x_3 + 5x_4 &= 0 \end{align*}
\begin{align*} x_1 - 2x_2 + x_3 - x_4 &= 0\\ 2x_1 - 4x_2 + x_3 + x_4 &= 0\\ x_1 - 2x_2 - 2x_3 + 3x_4 &= 0 \end{align*}
\begin{align*} x_1 - 2x_2 + x_3 - x_4 &= 0\\ x_1 + x_2 + x_3 - x_4 &= 0\\ x_1 \quad + x_3 - x_4 &= 0 \end{align*}
For Exercises C25-C27, solve the given homogeneous linear system. Compare your results to the results of the corresponding exercise in Section TSS.
\begin{align*} x_1 + 2x_2 + 3x_3 &= 0\\ 2x_1 - x_2 + x_3 &= 0\\ 3x_1 + x_2 + x_3 &= 0\\ \quad x_2 + 2x_3 &= 0 \end{align*}
\begin{align*} x_1 + 2x_2 + 3x_3 &= 0\\ 2x_1 - x_2 + x_3 &= 0\\ 3x_1 + x_2 + x_3 &= 0\\ \quad 5x_2 + 2x_3 &= 0 \end{align*}
\begin{align*} x_1 + 2x_2 + 3x_3 &= 0\\ 2x_1 - x_2 + x_3 &= 0\\ x_1 - 8x_2 - 7x_3 &= 0\\ \quad x_2 + x_3 &= 0 \end{align*}
Compute the null space of the matrix $A$, $\nsp{A}$. \begin{equation*} A= \begin{bmatrix} 2 & 4 & 1 & 3 & 8 \\ -1 & -2 & -1 & -1 & 1 \\ 2 & 4 & 0 & -3 & 4 \\ 2 & 4 & -1 & -7 & 4 \end{bmatrix} \end{equation*}
Find the null space of the matrix $B$, $\nsp{B}$. \begin{align*} B&= \begin{bmatrix} -6 & 4 & -36 & 6 \\ 2 & -1 & 10 & -1 \\ -3 & 2 & -18 & 3 \end{bmatrix} \end{align*}
Without doing any computations, and without examining any solutions, say as much as possible about the form of the solution set for corresponding homogeneous system of equations of each archetype that is a system of equations.
Archetype A,
Archetype B,
Archetype C,
Archetype D/Archetype E,
Archetype F,
Archetype G/Archetype H,
Archetype I,
Archetype J
For Exercises M50–M52 say as much as possible about each system's solution set. Be sure to make it clear which theorems you are using to reach your conclusions.
Prove or disprove: A system of linear equations is homogeneous if and only if the system has the zero vector as a solution.
Suppose that two systems of linear equations are equivalent. Prove that if the first system is homogeneous, then the second system is homogeneous. Notice that this will allow us to conclude that two equivalent systems are either both homogeneous or both not homogeneous.
Give an alternate proof of Theorem HSC that uses Theorem RCLS.
Consider the homogeneous system of linear equations $\homosystem{A}$, and suppose that $\vect{u}=\colvector{u_1\\u_2\\u_3\\\vdots\\u_n}$ is one solution to the system of equations. Prove that $\vect{v}=\colvector{4u_1\\4u_2\\4u_3\\\vdots\\4u_n}$ is also a solution to $\homosystem{A}$.