正交组

Introduction

Orthogonal Sets and the Projection Formula

The Gram–Schmidt Process

Two Methods to Compute the Projection

Which of the following sets of two vectors is an orthogonal set?

The set $\mathcal{B} = \{(1,1,1),\;(1,-1,0)\}$ is orthogonal but not orthonormal. How do you turn it into an orthonormal set spanning the same plane?

In one word: a set of nonzero vectors $\{u_1,\ldots,u_m\}$ with $u_i\cdot u_j = 0$ for all $i \neq j$ is called an ______ set.

Run one step of Gram-Schmidt on $v_1 = (1,1,0)$ and $v_2 = (1,1,1)$: set $u_1 = v_1$, then $u_2 = v_2 - \dfrac{v_2\cdot u_1}{u_1\cdot u_1}\,u_1$. Enter $u_2$ as a comma-separated triple, for example 1,-2,4.

Run Gram-Schmidt on $a = (1,-1,0,0)$, $b = (0,1,-1,0)$, $c = (0,0,1,-1)$: take $u_1 = a$, $u_2 = b - \dfrac{b\cdot u_1}{u_1\cdot u_1}\,u_1$, and $u_3 = c - \dfrac{c\cdot u_1}{u_1\cdot u_1}\,u_1 - \dfrac{c\cdot u_2}{u_2\cdot u_2}\,u_2$. Enter the third component of $u_3$ as a fraction, for example 3/4.

Let $\{u_1,\ldots,u_m\}$ be a set of nonzero vectors in $\mathbb{R}^n$ with $u_i\cdot u_j = 0$ whenever $i \neq j$. What follows?

Turn $u = (3,4)$ into a unit vector by dividing by its length. Enter the resulting unit vector as $x,y$ (fractions welcome, as in $a/b$).