Mathematics for Machine Learning: Linear Algebra

1. Introduction to Linear Algebra and to Mathematics for Machine Learning

The relationship between machine learning, linear algebra, and vectors and matrices

• 格拉姆-施密特正交化

$$\boldsymbol{\beta}_1 = \boldsymbol{v}_1,\quad \boldsymbol{\eta}_1 = {\boldsymbol{\beta}_1 \over \|\boldsymbol{\beta}_1\|}$$ $$\boldsymbol{\beta}_2 = \boldsymbol{v}_2-\langle \boldsymbol{v}_2, \boldsymbol{\eta}_1 \rangle \boldsymbol{\eta}_1, \quad \boldsymbol{\eta}_2 = {\boldsymbol{\beta}_2 \over \|\boldsymbol{\beta}_2\|}$$ $$\boldsymbol{\beta}_3 = \boldsymbol{v}_3 - \langle \boldsymbol{v}_3, \boldsymbol{\eta}_1 \rangle \boldsymbol{\eta}_1 - \langle \boldsymbol{v}_3, \boldsymbol{\eta}_2 \rangle \boldsymbol{\eta}_2 ,\quad \boldsymbol{\eta}_3 = {\boldsymbol{\beta}_3 \over \|\boldsymbol{\beta}_3\|}$$ $$\vdots$$ $$\boldsymbol{\beta}_n = \boldsymbol{v}_n -$$ $$\sum_{i=1}^{n-1}\langle \boldsymbol{v}_n, \boldsymbol{\eta}_i \rangle \boldsymbol{\eta}_i,\quad \boldsymbol{\eta}_n = {\boldsymbol{\beta}_n\over\|\boldsymbol{\beta}_n\|}$$