Vector Introduction (R^2 and R^n)

The easiest way to think about a vector is just a column of numbers. For example:

$$\vec{v} = \begin{bmatrix} 1 \\ 2 \\ \end{bmatrix} \, ; \, \vec{v} = \begin{bmatrix} 1\\ 2 \\ 3 \end{bmatrix} \, ; \, \vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \cdot \\ \cdot \\ \cdot \\ v_n \end{bmatrix} \, , \, v_1, ..., v_n \in \mathbb{R}$$

defines three different vectors. The number of entries / rows in the column is the dimension of the vector. We denote vectors by putting a little arrow on top of them e.g. $\vec{v}$ above.

When the dimension of the vector is 2, we can visualize it nicely on the Cartesian plane. Consider the vector:

$$\vec{v} = \begin{bmatrix} x \\ y \\ \end{bmatrix}$$

Then:

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