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Let's consider the following system of equations ... $$\begin{array}{c} x_1 + 8x_2 -4x_4 = 42 \\ x_2 + 2x_3 + 12x_4 = 8 \end{array}$$ and its compact representation ($Ax = b$) ... $$\begin{bmatrix} 1 & 0 & 8 & -4 \\ 0 & 1 & 2 & 12 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 42 \\ 8 \end{bmatrix}$$ The general solution of that system is ...

$$\left\{ x \in \mathbb{R}^4: \begin{bmatrix} 42 \\ 8 \\ 0 \\ 0 \end{bmatrix} + \lambda_1 \begin{bmatrix} 8 \\ 2 \\ -1 \\ 0 \end{bmatrix} + \lambda_2 \begin{bmatrix} -4 \\ 12 \\ 0 \\ -1 \end{bmatrix} , \lambda_1, \lambda_2 \in \mathbb{R} \right\}$$

Reference: 'Mathematics for Machine Learning' Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong

The vector $b$ can be graphically represented by ...

Vector graphically represented

and the particular solution can also be graphically represented by ...

Particular solution graphically represented

The general solution includes the scalars ($\lambda_1, \lambda_2$) therefore, it is possible to see graphically, the case where $\lambda_1$ and $\lambda_2$ are equal to 1 ...

General solution graphically represented

Still in the general solution, it is possible to change the value of the two scalars and see graphically the result ...

General solution graphically represented

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