**Example:** Scalar multiplication in \(\mathbb{R}^2\).

Let \(\mathbf{v} = (2, 1) \in \mathbb{R}^2\). Consider several scalar multiples:

## Positive scalars

- \(2 \cdot \mathbf{v} = 2 \cdot (2, 1) = (4, 2)\) — doubles the length, same direction
- \(\frac{1}{2} \cdot \mathbf{v} = (1, 0.5)\) — halves the length, same direction
- \(3 \cdot \mathbf{v} = (6, 3)\)

## Negative scalars

- \(-1 \cdot \mathbf{v} = (-2, -1)\) — same length, opposite direction
- \(-2 \cdot \mathbf{v} = (-4, -2)\) — doubles the length, opposite direction

## Zero scalar

- \(0 \cdot \mathbf{v} = (0, 0) = \mathbf{0}\) — the zero vector

## Geometric interpretation

All scalar multiples of \(\mathbf{v} = (2, 1)\) lie on the line through the origin with slope \(1/2\). The scalar \(a\) determines:
- **Magnitude:** \(\|a\mathbf{v}\| = |a| \cdot \|\mathbf{v}\|\)
- **Direction:** same as \(\mathbf{v}\) if \(a > 0\), opposite if \(a < 0\)

The set \(\{a \cdot \mathbf{v} \mid a \in \mathbb{R}\}\) is the **span** of \(\mathbf{v}\), a 1-dimensional subspace of \(\mathbb{R}^2\).