MATH 415 Study Guide - Final Guide: Row And Column Spaces, Linear Map

• School: University of Illinois
• Department: Mathematics
• Course: MATH 415
• Professor: J Li

Document Summary

Strang lecture: lecture 14: orthogonal vectors and subspaces. We will discuss later approximate solutions to ax = b. The underlying idea is that of projection, which we start on in this lecture. We need: recall: if v1, v2, . , vn are orthogonal (and non zero) they are indepen- dent: recall: to calculate coordinates for orthogonal vectors is easy: this uses v1 (c1v1 + c2v2 + + cnvn) = c1v1 v1. , vp form an orthogonal basis of v rn, w v , then w = c1v1 + c2v2 + + cpvp, with ci = vi w vi vi. This formula makes it easy to calculate the coordinates of w with respect to an orthogonal basis. Recall that an orthogonal basis v1, v2, . , vp is called orthonormal if all basis vectors are unit vectors: vi vi = 1. , vp is orthonormal then ci = vi w. The easy formula for the coordinates only works for orthogonal bases.