LAHW#14

1. LAHW#14 •Due December 27, 2010

2. 7.1 Inner-Product Spaces • 8. • In an inner-product space, let ||x|| = 7 and ||y|| = 9. How large can ||x - y|| be?

3. 7.1 Inner-Product Spaces • 9. • Determine whether this mapping is linear:(Here v is any fixed vector, and the setting can be any inner-product space.)

4. 7.1 Inner-Product Spaces • 10. • For the matrixfind a simple description of the null space and the orthogonal complement of the row space.

5. 7.1 Inner-Product Spaces • 11. • Answer these questions and draw simple sketches to illustrate them:a. If x⊥y, does it follow that for all scalars α and β, we have αx⊥βy.b. If x⊥y and y⊥z, does it follow that x⊥z?c. If x⊥y and x⊥z, does it follow that αx⊥(βy+γz )for all scalars α, β, and γ?

6. 7.1 Inner-Product Spaces • 12. • Explain this apparent contradiction:For the vector x = (7 + 3i, -3 + 7i), we find that ||x||2 = <x, x> = x12 + x22 = (7 + 3i)2 + (-3 + 7i)2 = (49 + 42i - 9) + (9 – 42i - 40) = 0.

7. 7.1 Inner-Product Spaces • 13. • Find the orthogonal projection of x = (2, 2, 3) onto the subspace of R3 spanned by two vectors u = (4, 1, -2) and v = (1, -2, 1). Note that u⊥v. Check your work by verifying independently the two properties that the projection should have.

8. 7.1 Inner-Product Spaces • 31. • Let A be a square matrix whose columns form an orthogonal set. If we normalize each row of A, we obtain a new matrix, B. Is B an orthogonal matrix? What interesting properties does B have? Answer the same questions for the matrix C obtained by normalizing the columns of A.

9. 7.1 Inner-Product Spaces • 88. • Define an inner product in R3 by the equation <x, y>o = 3x1y1 + 5x2y2 + 2x3y3. (The subscript o us there to remind us that this is not the standard inner product.) What is the largest value that <x, y>o can attain when y is fixed vector and x is a free vector constrained only by <x, x>o ≦ 7.