wksht13 - MATH 152 L1 & L2 Applied Linear Algebra and...

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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Week 13 Worksheet (May 4, 2005): Orthogonality Spring 2004-05 Q1. (Orthogonal set) Find k so that u = (1, 2, k, 3) and v = (3, k, 7, −5) in R4 are orthogonal. Q2. (Orthogonal set) Let v1 = (1, 4, 1), v2 = (−2, 1, −2), v3 = (1, 0, −1). Do v1 , v2 , v3 form an orthogonal set? If yes, rescale the vectors to make the them orthonormal. Q3. (Orthogonal set) Find a nonzero vector w that is orthogonal to v1 = (1, 2, 1), v2 = (2, 5, 4) in R3 . Q4. (Orthogonal basis) Let v1 = (1, 1, 0, −1), v2 = (1, 2, 1, 3), v3 = (1, 1, −9, 2), v4 = (16, −13, 1, 3). Show that the vectors v1 , v2 , v3 , v4 form an orthogonal set and they indeed form a basis of R4 . Q5. (Orthogonal basis) Show that v1 = (0, 1, 0), v2 = (1, 0, 1) are orthogonal. Find a third vector v3 so that v1 , v2 , v3 form an orthogonal basis of R3 . Q6. (Orthogonal basis) Find an example of 3 × 3 matrix with orthogonal column vectors but nonorthogonal row vectors. √ 1/ 2 Q7. (Orthogonal matrix) Find all orthogonal matrices U of the form 0 √ 1/ 2 x √ 1/ 2 y a b . c Q8. (Non-orthogonal matrix) matrix. Find an example of invertible 2 × 2 matrix that is not an orthogonal Q9. (True or false) Suppose U and V are square matrices. (a) If U and V are orthogonal matrices, then so is UV. (b) If U is an orthogonal matrix, then so is −U. (c) If UUt is diagonal, then the columns of U are orthogonal. ...
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This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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wksht13 - MATH 152 L1 & L2 Applied Linear Algebra and...

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