PREFACE

• how to use matrices to the simultaneous equations
• \end{bmatrix}
• \end{bmatrix}
  • Row Picture
  • Column Picture
• how about three elements
• A
• b
  • Row Picture
  • Column picture
• question about the existence of every b

how to use matrices to the simultaneous equations

$$\begin{cases} 2x - y &= 0, \\ -x + 2y &= 3. \end{cases}$$

What is the coefficient matrix?

$$A \cdot B = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \cdot \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}$$

Hence

$$\begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix}$$

It’s called

$$Ax=b$$

Row Picture

Column Picture

$$x \cdot \begin{bmatrix} 2 \\ -1 \end{bmatrix} + y \cdot \begin{bmatrix} -1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix}$$

This is the linear combination of columns.

how about three elements

Just simply combine the x and y, we would gain $x=1 \quad and \quad y=2$.

$$\begin{cases} 2x - y &= 0, \\ -x + 2y - z &= -1 \\ -3y + 4z &= 4 \end{cases}$$ $$A = \begin{bmatrix} 2 & -1 & 0\\ -1 & 2 & -1\\ 0 & -3 & 4 \end{bmatrix}$$ $$b = \begin{bmatrix} 0\\ -1\\ 4 \end{bmatrix}$$

Hence

$$\begin{bmatrix} 2 & -1 & 0\\ -1 & 2 & -1\\ 0 & -3 & 4 \end{bmatrix} \cdot \begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} 0\\ -1\\ 4 \end{bmatrix}$$

Row Picture

seperately let two of the elements be equal to zero, then get the third element’s coordinate.

Column picture

$$x \cdot \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} + y \cdot \begin{bmatrix} -1 \\ 2 \\ -3 \end{bmatrix} + z \cdot \begin{bmatrix} 0 \\ -1 \\ 4 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \\ 4 \end{bmatrix}$$

These equations ask us to combine the three vectors in the coordinate. Eventually we got x=0 y=0 z=1

question about the existence of every b

Can I solve Ax=b for every b?

Do the linear combs of the columns fill 3-D space?

For the matrix, the answer is YES. But if these b forming in the same plane, you will get the same plain as the result.

---

Welcome to point out the mistakes and faults!