11-coordinates

11-coordinates - COORDINATES Math 21b, O. Knill HOMEWORK:...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: COORDINATES Math 21b, O. Knill HOMEWORK: Section 3.4: 2,14,16,22,26,32*,38* B-COORDINATES. Given a basis ~v 1 , . . . ~v n , define the matrix S = | . . . | ~v 1 . . . ~v n | . . . | . It is invertible. If ~x = i c i ~v i , then c i are called the B-coordinates of ~v . We write [ ~x ] B = c 1 . . . c n . If ~x = x 1 . . . x n , we have ~x = S ([ ~x ] B ). B-coordinates of ~x are obtained by applying S- 1 to the coordinates of the standard basis: [ ~x ] B = S- 1 ( ~x ) EXAMPLE. If ~v 1 = 1 2 and ~v 2 = 3 5 , then S = 1 3 2 5 . A vector ~v = 6 9 has the coordinates S- 1 ~v =- 5 3 2- 1 6 9 =- 3 3 Indeed, as we can check,- 3 ~v 1 + 3 ~v 2 = ~v . EXAMPLE. Let V be the plane x + y- z = 1. Find a basis, in which every vector in the plane has the form a b . SOLUTION. Find a basis, such that two vectors v 1 , v 2 are in the plane and such that a third vector v 3 is linearly independent to the first two. Since (1 , , 1) , (0 , 1 , 1) are points in the plane and (0 , , 0) is in the plane, we can choose ~v 1 = 1 1 ~v 2 = 1 1 and ~v 3 = 1 1- 1 which is perpendicular to the plane....
View Full Document

Ask a homework question - tutors are online