{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# sol56 - Selected Solutions Leon 5.6 25 1 10 is upper...

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

Selected Solutions, Leon § 5.6 5.6.2 (b) Factor A = 2 5 1 10 into a product QR , where Q is an orthogonal matrix and R is upper triangular. Solution : We use the modified Gram-Schmidt algorithm. Let a 1 , a 2 denote the columns of A . We begin by finding r 11 = k a 1 k = 5, and normalizing a 1 to obtain q 1 = ( 2 5 , 1 5 ) T . Next, we find r 12 = q T 1 a 2 = 4 5. We then update a 2 by subtracting from a 2 its projection onto q 1 , i.e., a 2 := a 2 - r 12 q 1 = ( - 3 , 6) T . This completes one iteration of the algorithm. We now calculate r 22 = k a 2 k = 3 5, and normalize a 2 to obtain q 2 = ( - 1 5 , 2 5 ) T . Assembling these pieces, we have A = 2 5 1 10 = 2 / 5 - 1 / 5 1 / 5 2 / 5 5 4 5 0 3 5 = QR. 5.6.3 Given the basis { (1 , 2 , - 2) T , (4 , 3 , 2) T , (1 , 2 , 1) T } for R 3 , use the Gram-Schmidt process to obtain an orthonormal basis. Solution : We use the modified Gram-Schmidt process, which is a better mousetrap. We simply discard the matrix R , retaining the columns of Q as our orthonormal basis.

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 / 2

sol56 - Selected Solutions Leon 5.6 25 1 10 is upper...

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

View Full Document
Ask a homework question - tutors are online