🧭 Parametric vs Argument View in Linear Systems

🎯 Goal

Understand how the same solution can be written in two different styles:

• Row form: each variable is written separately.
• Vector form: the whole solution is written as a single vector made from parts.

🧮 Example

Given:

🧾 Row Form — View by Parameter

Each variable is written on its own line:

• $x = -4t - 3s$
• $y = t$
• $z = s$

🔍 This shows how each variable depends on $t$ and $s$.

📦 Vector Form — View by Argument

$$ \begin{bmatrix} x \ y \ z \end{bmatrix} = t \begin{bmatrix} -4 \ 1 \ 0 \end{bmatrix}

• s \begin{bmatrix} -3 \ 0 \ 1 \end{bmatrix} $$

🔍 This shows how the whole solution is built by combining two fixed vectors using $t$ and $s$.

🧠 Summary

🪄 Analogy

• Row form: Like writing a recipe for each ingredient.
• Vector form: Like mixing two colors to get a final shade.