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

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

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 = ~

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

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

{[ snackBarMessage ]}

### 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
Ask a homework question - tutors are online