Exercise 1 - Homogeneous Linear System¶
Consider the homogeneous linear system:
$$\begin{align*} x + 2y + z + 3w &= 0 \\ x -\ y\ \ \ \ \ \ \ +\ \ w &= 0 \\ y - z + 2w &= 0 \end{align*}$$
One solution to the homogeneous system is the vector $\begin{bmatrix} -2 \\ -1 \\ \ \ 1 \\ \ \ 1 \end{bmatrix}$.
Explain why $\begin{bmatrix}\ \ 4 \\ \ \ 2 \\ -2 \\ -2 \end{bmatrix}$ is also a solution.
Exercise 2 - Homogeneous Linear System¶
Let vectors $\mathbf{x}_1$ and $\mathbf{x}_2$ be solutions of the homogeneous linear system $A\mathbf{x} = \mathbf{0}$. Explain why the vector $2\mathbf{x}_1 - 3\mathbf{x}_2$ is also a solution.
Exercise 3 - Two Linear System¶
Consider the two linear systems represented by the augmented matrices
$$ \left[ \begin{array}{ccc|c} 1 &\ \ 1 & -5 &\ \ 3 \\ 1 &\ \ 0 & -2 &\ \ 1 \\ 2 & -1 & -1 &\ \ 0 \end{array} \right],\ \left[ \begin{array}{ccc|c} 1 &\ \ 1 & -5 & -9 \\ 1 &\ \ 0 & -2 & -3 \\ 2 & -1 & -1 &\ \ 0 \end{array} \right] $$
If the first system is consistent, explain why the second system is also consistent.
Exercise 4 - Homogeneous Linear System¶
The vectors $\mathbf{x}_1$ and $\mathbf{x}_2$ are solutions to the linear system $A\mathbf{x} = \mathbf{b}$. Is the vector $2\mathbf{x}_1 - 3\mathbf{x}_2$ also a solution? Explain your answer.
Exercise 5 - Combination of Linear Systems¶
The linear systems $A\mathbf{x} = \mathbf{b}_1$ and $A\mathbf{x} = \mathbf{b}_2$ are consistent. Is the system $A\mathbf{x} = \mathbf{b}_1 + \mathbf{b}_2$ necessarily consistent? Explain your answer.