>[!summary] Vectors in linear algebra are denoted by a scalar point for each available axis. > > **Key equations:** > **For the following equations, it's assumed that vectors are in $\mathbb{R^2}$ but is valid for any $\mathbb{R^n}$** > They are equal if: >$\begin{array}{c} \text{We say} \space {\begin{bmatrix} x_1 \\ x_2\end{bmatrix}} = {\begin{bmatrix} y_1 \\ y_2\end{bmatrix}} \space \text{if:} \\ x_1 = y_1 \\ x_2 = y_2 \end{array} >$ We generally add vectors in this way: > >$\begin{array}{c} \vec{x} + \vec{y} = {\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}} + {\begin{bmatrix} y_1 \\ y_2 \end{bmatrix}} = {\begin{bmatrix} x_1 + y_1 \\ x_y + y_2 \end{bmatrix}}\end{array}$ And if t is a scalar multiple then multiply vectors are in this way: $t\vec{x} = t{\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}} = {\begin{bmatrix} tx_1 \\ tx_2 \end{bmatrix}}$ > As well: >$\begin{array}{c} \text{If $v_1...v_k \in \mathbb{R^2}$ and $c_1... c_k \in \mathbb{R}$} \\ \text{Then the sum $c_1v_1 ... c_k v_k$ is a linear combination of $v_1...v_k$} \end{array}$ >[!info]+ Read Time **⏱ 3 mins** # Defining A Set of Vectors Given a graph with two axes ($\mathbb{R}^2$) it's common to denote a point P by the distance across the x-axis and distance across the y-axis. For example, a point ($x_{1},x_{2}$) in linear algebra is denoted as ${\begin{bmatrix} x_1 \\ x_2\end{bmatrix}}$. ![[ve_1.png|500]] > [!note] Explanation A point $(x_1,x_{2})$ has a displacement in the x-axis by $x_{1}$ and displacement in the y-axis by $x_{2}$ ## Generalization The generalized for a set a vector can work in an $\mathbb{R^n}$ but well generalize this for a function in $\mathbb{R^2}$ $\begin{array}{c} \mathbb{R} ^2 = {\begin{bmatrix} x_1 \\ x_2\end{bmatrix}} | x_1, x_2 \in \mathbb{R} \\ \\ \text{We say} \space {\begin{bmatrix} x_1 \\ x_2\end{bmatrix}} = {\begin{bmatrix} y_1 \\ y_2\end{bmatrix}} \space \text{if:} \\ x_1 = y_1 \\ x_2 = y_2 \end{array}$ # Proving Vector Addition & Multiplication >[!warning] Assumption For this derivation, assume an example in physics, where two forces are multiplied or added together, and then generalize the solutions. Suppose two forces $F_1$ and $F_2$. Where $F_{1}=150N$ in the x-axis and $F_{2}=150N$ in the y direction. The force when $F_{1}+F_{2}$ is equal to the force in the x-axis and the force in the y-axis. If a force were multiplied, that would be adding a multiplicative scalar amount to a force. ![[Probjv (1).png]] $\begin{array}{c} F_1 = {\begin{bmatrix} 150 \\ 0\end{bmatrix}} \\ F_2 = {\begin{bmatrix} 0 \\ 150 \end{bmatrix}} \\ F_1 + F_2 = {\begin{bmatrix} 150 \\ 0\end{bmatrix}} + {\begin{bmatrix} 0 \\ 150 \end{bmatrix}} = {\begin{bmatrix} 150 \\ 150 \end{bmatrix}} \\ 2F_1 = {\begin{bmatrix} 300 \\ 0 \end{bmatrix}} \end{array}$ ## Generalization For generalizing addition and multiplication from this example we assume its in $\mathbb{R^2}$ but it works for any value of $\mathbb{R^n}$ $\begin{array}{c} \text{Let $\vec{x} = {\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}} $ and $\vec{y} = {\begin{bmatrix} y_1 \\ y_2\end{bmatrix}} $} \\ \vec{x} + \vec{y} = {\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}} + {\begin{bmatrix} y_1 \\ y_2 \end{bmatrix}} = {\begin{bmatrix} x_1 + y_1 \\ x_y + y_2 \end{bmatrix}} \\\\ \text{If we define t as a sclar multiple then} \\ t\vec{x} = t{\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}} = {\begin{bmatrix} tx_1 \\ tx_2 \end{bmatrix}} \end{array}$ As well, we can make a definition from the example assuming that we a vector in $\mathbb{R^2}$ and a scalar multiple in $\mathbb{R}$. From the example for any form of vectors we make the definition: $\begin{array}{c} \text{If $v_1...v_k \in \mathbb{R^2}$ and $c_1... c_k \in \mathbb{R}$} \\ \text{Then the sum $c_1v_1 ... c_k v_k$ is a linear combination of $v_1...v_k$} \end{array}$ --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidan-MathMatter) with a star and help others learn more easily. ---