Definition: The scalar multiplication of a vector (\mathbf{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{F}^n) by a scalar (a \in \mathbb{F}) is the vector obtained by multiplying each component by (a):
[a \cdot \mathbf{v} = (a \cdot v_1,; a \cdot v_2,; \ldots,; a \cdot v_n)]
Component-wise formula:((a\mathbf{v})_j = a \cdot v_j) for each (j = 1, 2, \ldots, n).
Geometric interpretation ((\mathbb{R}^n)): Scalar multiplication scales the vector by (|a|) and, if (a < 0), reverses its direction.
Algebraic properties: Scalar multiplication satisfies distributivity over vector addition and scalar addition, associativity, and has a scalar identity: (1 \cdot \mathbf{v} = \mathbf{v}).
**Definition:** The *scalar multiplication* of a vector \(\mathbf{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{F}^n\) by a scalar \(a \in \mathbb{F}\) is the vector obtained by multiplying each component by \(a\):
\[a \cdot \mathbf{v} = (a \cdot v_1,\; a \cdot v_2,\; \ldots,\; a \cdot v_n)\]
**Component-wise formula:** \((a\mathbf{v})_j = a \cdot v_j\) for each \(j = 1, 2, \ldots, n\).
**Examples:**
- In \(\mathbb{R}^3\): \(2 \cdot (1, 3, -1) = (2, 6, -2)\)
- In \(\mathbb{C}^2\): \(i \cdot (1+i, 2) = (i-1, 2i)\)
- With negative scalar: \(-1 \cdot (3, -2, 5) = (-3, 2, -5)\)
**Geometric interpretation (\(\mathbb{R}^n\)):** Scalar multiplication scales the vector by \(|a|\) and, if \(a < 0\), reverses its direction.
**Algebraic properties:** Scalar multiplication satisfies distributivity over vector addition and scalar addition, associativity, and has a scalar identity: \(1 \cdot \mathbf{v} = \mathbf{v}\).