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解集

1. Introduction

Introduction

2. Homogeneous Systems

Homogeneous Systems

3. Inhomogeneous Systems

Inhomogeneous Systems

4. Solution Sets and Column Spans

Solution Sets and Column Spans

5. A free variable in $Ax=0$

Suppose the homogeneous equation $Ax=0$ has a free variable. Which of the following must be true?

6. Can it be nontrivial?

Let $A=\begin{pmatrix} 1 & 3 & 4 \\ 2 & -1 & 2 \\ 1 & 0 & 1 \end{pmatrix}$ . Does the homogeneous equation $Ax=0$ have a nontrivial solution?

7. Vocabulary: trivial solution

The solution $x=0$ of a homogeneous system $Ax=0$ is called the ______ solution.

8. Parametric form of a plane

For $A = \begin{pmatrix} 1 & -1 & 2 \\ -2 & 2 & -4 \end{pmatrix}$ , the solution set of $Ax=0$ is

x = x_2\begin{pmatrix}1\\1\\0\end{pmatrix} + x_3\begin{pmatrix}\ ?\ \\0\\1\end{pmatrix}.

Enter the first component of the vector multiplying $x_3$ (a number).

9. Special solutions: inhomogeneous case

The matrix equation $Ax = b$ has reduced row echelon form

\left(\begin{array}{cccc|c} 1 & 0 & -3 & 2 & 6 \\ 0 & 1 & 5 & -1 & -10 \end{array}\right),

with free variables $x_3,x_4$ . Write the complete solution in the form $x = x_p + x_3v_3 + x_4v_4$ . Enter the first component of $x_p$ (a number).

10. Structure of a solution set

The vector $p$ is one solution of $Ax = b$ , and the solutions of $Ax = 0$ are exactly $\operatorname{Span}\{u\}$ . Which set lists all solutions of $Ax = b$ ?

11. Walk along a translated line

Solutions of $Ax = b$ have the form $(1,0) + t\,(2,1)$ where $t$ is any real number. Enter the solution that corresponds to $t = 2$ , as $x,y$ .

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