# Complete Solution to Ax = b

Korey Hinton -> Blog -> Math -> Linear Algebra -> Complete Solution to Ax = b

## Finding complete solution

1. Reduce augmented matrix to Reduced Row Echelon Form
2. Find particular solution
3. Find special (nullspace) solutions in terms of free variables
4. Combine particular solution with nullspace solutions

### Example

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

#### 1) Reduce augmented matrix to Reduced Row Echelon Form

\begin{bmatrix} 1 & 1 & 3 & : & 1\\ 2 & 1 & 1 & : & 1\\ 3 & 2 & 4 & : & 2\\ \end{bmatrix}

-2R1 + R2, -3R1 + R3, -1R2, 2R2 + R3, -R2 + R1

\begin{bmatrix} 1 & 0 & -2 & : & 0\\ 0 & 1 & 5 & : & 1\\ 0 & 0 & 0 & : & 0\\ \end{bmatrix}

The pivot variables are x and y and the free variable is z. To be a pivot variable it's column position on the right-hand side (so x has position 1) should have a 1 in the left-hand side column & row position (x has col 1 row 1) and anything else in that column must be a zero.

#### 2) Find particular solution

Let the free variable = 0. Thus 1x + 0y + -2*(0) = 0, so x = 0. And 0x + 1y +5*(0) = 1. So y = 1. $x_{p} = \begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix}$

#### 3) Find special (nullspace) solutions in terms of free variables

Use augmented matrix with the right-hand side = 0 to calculate the nullspace N(A): $\begin{bmatrix} 1 & 0 & -2 & : & 0\\ 0 & 1 & 5 & : & 0\\ 0 & 0 & 0 & : & 0\\ \end{bmatrix}$ Solve for x,y in terms of z.

x - 2z = 0

y + 5z = 0

x = 2z

y = -5z

$s_{1} = z \begin{bmatrix} 2\\ -5\\ 1\\ \end{bmatrix}$

#### 4) Combine particular solution with nullspace solutions

Complete solution to Ax = b is xp + xn.

xn = s1 + … sn.

So the complete solution for this example is:

$\begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix} + z \begin{bmatrix} 2\\ -5\\ 1\\ \end{bmatrix}$

Date: 2014-06-28T04:59+0000

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