MATH 415 Study Guide - Final Guide: Orthogonal Complement, Standard Basis, Linear Map

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

Document Summary

Suggested practice exercises: ch 3. 2:17, 18, 24; 3. 4: 2, 3. 1. 1 last time: orthogonal projection of x onto y: xy = x y y y y x y. Error x = x xy is orthogonal to y: if y1, . , yn is an orthogonal basis of v , and x is in v , then x = c1y1 + + cnyn with cj = x yj yj yj x xy. If v is any vector, then the projection matrix p for projection onto v, in standard coordinates, is given by. Remark. x decomposes as the sum of its projections onto each vector in the orthogonal basis. The formulas simplify when you project on unit vectors: all denom- inators are then 1. Then, each x in rn can be uniquely written as. |{z} x x v1 xw v2 xw is the orthogonal projection of x onto w . x x v1 xw v2.