c6s2 - 6.2 The Gram-Schmidt Process 1 Chapter 6....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Chapter 6. Orthogonality 6.2 The Gram Schmidt Process Defnition. Aset { ~v 1 ,~v 2 ,...,~v k } of nonzero vectors in R n is orthogonal if the vectors ~v j are mutually perpendicular — that is, if ~v i · ~v j =0fo r i 6 = j . Theorem 6.2. Orthogonal Bases. Let { ~v 1 ,~v 2 ,...,~v k } be an orthogonal set of nonzero vectors in R n .Th en this set is independent and consequently is a basis for the subspace sp( ~v 1 ,~v 2 , ...,~v k ) . ProoF. Let j be an integer between 2 and k .Con s id e r ~v j = s 1 ~v 1 + s 2 ~v 2 + ··· + s j 1 ~v j 1 . If we take the dot product of each side of this equation with ~v j then, since the set of vectors is orthogonal, we get ~v j · ~v j = 0, which contradicts the hypothesis that ~v j 6 = ~
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

c6s2 - 6.2 The Gram-Schmidt Process 1 Chapter 6....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online