Gram-Schmidt Process

Let $X$ be a set of vectors that form a basis for subspace $W$ of ${\mathbb{R}}^n$, Let $V$ be a set of vectors that form an Orthogonal Basis for $W$ The Gram-Schmidt process defines how $V$ can be derived from $X$ It depends on Orthogonal Decomposition

If $X=\{\overrightarrow{x_1},\cdots \overrightarrow{x_p}\}$, iteratively define vectors in $V=\{\overrightarrow{v_1},\cdots ,{\vec{v}}_p\}$ $\overrightarrow{v_1}=\overrightarrow{x_1}$ $\overrightarrow{v_2}=\overrightarrow{x_2}-pro{j}_{\overrightarrow{v_1}}\overrightarrow{x_2}$ $\overrightarrow{v_3}=\overrightarrow{x_3}-pro{j}_{\overrightarrow{v_1}}\overrightarrow{x_3}-pro{j}_{\overrightarrow{v_2}}\overrightarrow{x_3}$ $\cdots$ ${\vec{v}}_p=\overrightarrow{x_p}-pro{j}_{{\vec{v}}_1}\overrightarrow{x_p}-\cdots -pro{j}_{{\vec{v}}_{p-1}}{\vec{x}}_p$

$${\vec{v}}_p=\overrightarrow{x_p}-\sum _{i=1}^{p-1}{\text{proj}}_{{\vec{v}}_i}{\vec{x}}_p$$

Examples

HW 6.4 Q1 HW 6.4 Q2