Created almost 9 years ago, updated 5 months ago
ID23795
Definition: The geometric properties of (\mathbb{R}^n) (or (\mathbb{C}^n)) arise from the dot product (inner product) and the norm it induces.
Dot product: For (\mathbf{u} = (u_1,\ldots,u_n)) and (\mathbf{v} = (v_1,\ldots,v_n)):
[\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i]
Norm (length): (|\mathbf{u}| = \sqrt{\mathbf{u} \cdot \mathbf{u}} = \sqrt{u_1^2 + \cdots + u_n^2})
Distance: (d(\mathbf{u}, \mathbf{v}) = |\mathbf{u} - \mathbf{v}|)
Key geometric concepts:
- Angle: (\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|})
- Orthogonality: (\mathbf{u} \perp \mathbf{v}) iff (\mathbf{u} \cdot \mathbf{v} = 0)
- Cauchy-Schwarz inequality: (|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}| |\mathbf{v}|)
- Triangle inequality: (|\mathbf{u} + \mathbf{v}| \leq |\mathbf{u}| + |\mathbf{v}|)
Example: In (\mathbb{R}^2), the vectors ((1,0)) and ((0,1)) are orthogonal since their dot product is (1 \cdot 0 + 0 \cdot 1 = 0).
**Definition:** The *geometric properties* of \(\mathbb{R}^n\) (or \(\mathbb{C}^n\)) arise from the dot product (inner product) and the norm it induces.
**Dot product:** For \(\mathbf{u} = (u_1,\ldots,u_n)\) and \(\mathbf{v} = (v_1,\ldots,v_n)\):
\[\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i\]
**Norm (length):** \(\|\mathbf{u}\| = \sqrt{\mathbf{u} \cdot \mathbf{u}} = \sqrt{u_1^2 + \cdots + u_n^2}\)
**Distance:** \(d(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\|\)
**Key geometric concepts:**
- **Angle:** \(\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}\)
- **Orthogonality:** \(\mathbf{u} \perp \mathbf{v}\) iff \(\mathbf{u} \cdot \mathbf{v} = 0\)
- **Cauchy-Schwarz inequality:** \(|\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\|\)
- **Triangle inequality:** \(\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|\)
**Example:** In \(\mathbb{R}^2\), the vectors \((1,0)\) and \((0,1)\) are orthogonal since their dot product is \(1 \cdot 0 + 0 \cdot 1 = 0\).