# Definition of vector-scalar multiplication

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# Parents
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* **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 \(a\):
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\[a \cdot \mathbf{v} = (a \cdot v_1,\; a \cdot v_2,\; \ldots,\; a \cdot v_n)\]
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**Component-wise formula:** \((a\mathbf{v})_j = a \cdot v_j\) for each \(j = 1, 2, \ldots, n\).
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**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: \(-1 \cdot (3, -2, 5) = (-3, 2, -5)\)
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**Geometric interpretation (\(\mathbb{R}^n\)):** Scalar multiplication scales the vector by \(|a|\) and terminology
* , if \(a < 0\), reverses its direction.
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**Algebraic properties:** Scalar multiplication satisfies distributivity over vector addition and scalar addition, associativity, and has a scalar identity: \(1 \cdot \mathbf{v} = \mathbf{v}\).
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# Parents
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* Algebraic properties of R^n (or C^n)
* Definition and terminology⏎