Math 511: Linear Algebra
Solutions of Linear Systems
Solutions of Linear Systems¶
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Write a paragraph to answer each question. Do not perform any calculations, but instead base your explanations on appropriate properties from the course.
1. One Solution of the homogeneous linear system¶
$$ \begin{align*} x + 2y + z + 3w &= 0 \\ x -\ y\ \ \ \ \ \ \ +\ \ w &= 0 \\ y - z + 2w &= 0 \end{align*} $$
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.
2. The vectors $\mathbf{x}_1$ and $\mathbf{x}_2$ are 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.¶
3. Consider the two linear systems represented by the augmented matrices¶
$$ \begin{bmatrix} 1 &\ \ 1 & -5 &\ \ 3 \\ 1 &\ \ 0 & -2 &\ \ 1 \\ 2 & -1 & -1 &\ \ 0 \end{bmatrix},\ \begin{bmatrix} 1 &\ \ 1 & -5 & -9 \\ 1 &\ \ 0 & -2 & -3 \\ 2 & -1 & -1 &\ \ 0 \end{bmatrix} $$
If the first system is consistent, explain why the second system is also consistent.