Vectors

Norm of a Vector

• (d12.1.3) - $\Vert \text{a}\Vert :=\sqrt{\text{a}\cdot \text{a}}=\sqrt{{\sum }_{i=1}^n{a}_i^2}$
• (q12.1.4) $\Vert \text{a}\Vert =0⟺\text{a}=\text{0}$
• Homogeneity (q12.1.5) $\Vert t\text{a}\Vert =|t|\cdot \Vert \text{a}\Vert$
• Cauchy–Schwarz inequality (12.1.4) $|\text{a}\cdot \text{b}|\le \Vert \text{a}\Vert \cdot \Vert \text{b}\Vert$
• (q12.1.7) $|\text{a}\cdot \text{b}|=\Vert \text{a}\Vert \cdot \Vert \text{b}\Vert ⟺\text{a},\text{b}$ are linearly inpendent
• (q12.1.8) Triangle inequality for vectors $\Vert \text{a}+\text{b}\Vert \le \Vert \text{a}\Vert +\Vert \text{b}\Vert$
• Parallelogram Equation for Vectors $\Vert \text{u}+\text{v}{\Vert }^2+\Vert \text{u}-\text{v}{\Vert }^2=2(\Vert \text{u}{\Vert }^2+\Vert \text{v}{\Vert }^2)$

Coordinate vector

• (8.4.4) Vectors set are linearly independent, if and only if, its coordinate vectors are linearly independent.
• (10.2.1) coordinate vector of image - $[T(v){]}_C=[T{]}_C^B[v{]}_B$

Orthogonality

• (d12.2.1) orthogonality of two vectors - $\text{a}$ and $\text{b}$ are called orthogonal if $\text{a}\cdot \text{b}=0$ (This relationship is denoted $\text{a}\perp \text{b}$)
• (q12.2.3) Generalized Theorem of Pythagoras: $\text{a}\perp \text{b}⟹\Vert \text{a}+\text{b}{\Vert }^2=\Vert \text{a}{\Vert }^2+{\Vert \text{b}\Vert }^2$
• (d12.2.2) $\mathbf{v}\perp U$ if for all vectors $\mathbf{u}\in U$, $\mathbf{v}\cdot \mathbf{u}=0$