Geometric properties of R^n (or C^n)

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).