Name: UIN: Problem Set 6 Math 423, Section 500, Spring 2015 Due: Friday, March 13. Review Sections 6C and 7A in your textbook. Complete the following items, staple this page to the front of your work, and turn your assignment in at the beginning of class on Friday, March 13. One of the problems is considered extra credit. Remember to fully justify all your answers, and provide complete details. Neatness is greatly appreciated. 1. Suppose v1 , . . . , vm ∈ V, where V is an inner product space. Prove that {v1 , . . . , vm }⊥ = (Span(v1 , . . . , vm ))⊥ . (The left hand side is the orthogonal complement of a subset of V, the right hand side is the orthogonal complement of a subspace of V.) 2. Suppose that U is a finite-dimensional subspace of an inner product space V. Prove that U ⊥ = {0} if and only if U = V. 3. Let V be a finite-dimensional inner product space. Let U be a subspace fo V with basis u1 , . . . , um , and suppose that u1 , . . . , um , w1 , . . . , wn is a basis of V. Apply the Gram–Schmidt procedure to this basis of V, to obtain e1 , . . . , em , f1 , . . . , fn . Prove that e1 , . . . , em is an orthonormal basis of U, and f1 , . . . , fn is an orthonormal basis of U ⊥ . 4. Let U = Span (1, 2, 3, −4), (−5, 4, 3, 2) ⊂ R4 . Consider R4 with the standard inner product. Find an orthonormal basis of U and an orthonormal basis of U ⊥ . 5. Suppose that V is a finite dimensional inner product space, and U is a subspace of V. Prove that PU ⊥ = idV − PU , where idV is the identity operator on V. 6. Suppose that T ∈ L (V) and U is a finite dimensional subspace of V (V is an inner product space). Prove that U is T -invariant if and only if PU T PU = T PU . 7. Suppose V is a finite dimensional inner product space, T ∈ L (V) and U is a subspace of V. Prove that U and U ⊥ are both T -invariant if and only if PU T = T PU . Page 1 8. Let V = R2 with the standard inner product, and W = Span (1, 2) . Consider the orthogonal projection PW . Find the matrix of PW with respect to the standard basis (1, 0), (0, 1) of R2 . 9. Let V = R3 with the standard inner product, and W = Span (1, 0, 1) . Consider the orthogonal projection PW . Find the matrix of PW with respect to the standard basis (1, 0, 0), (0, 1, 0), (0, 0, 1) of R3 . 10. Consider R4 with the standard inner product, and let U = Span (1, 1, 0, 0), (1, 1, 1, 2) . Find u ∈ U such that ku − (1, 2, 3, 4)k is as small as possible. 11. Find p ∈ P3 (R) such that p(0) = 0, p0 (0) = 0 and Z 1 |2 + 3x − p(x)|2 dx 0 is as small as possible. 1/10 2/10 3/10 4/10 5/10 6/10 7/10 8/10 9/10 10/10 11/10 Through the course of this assignment, I have followed the Aggie Code of Honor. An Aggie does not lie, cheat or steal or tolerate those who do. Signed: Page 2 Total/100