{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

601-hw-vii - MATH 601 AUTUMN 2008 Additional homework VII...

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

View Full Document Right Arrow Icon
MATH 601, AUTUMN 2008 Additional homework VII, November 19 PROBLEMS 1. Let V be an inner-product space of dimension n , and let v be a nonzero vector in V . Use the rank-nullity theorem to show that dim v = n - 1. 2. Let W be a subspace of an inner-product space V of dimension n , and let w 1 , . . . , w s be a basis of W . For the linear operator T : V K s given by Tv = ( v, w 1 , . . . , v, w s ), show that T restricted to W is an isomorphism W K s . Conclude from this that T : V K s is surjective, and then use the rank-nullity theorem to obtain a proof (different from the one shown in class) that dim W = dim V - dim W . 3. Using the standard (sesquilinear) Hermitian inner product , on C 4 , and letting V C 4 be the subspace consisting of all ( x, y, z, t ) , with x + z - it = x - iy - z = 0, write an explicit formula for the V and V components (relative to the decomposition C 4 = V V ) of each ( x, y, z, t ) C 4 . 4. An endomorphism T : V V is called an involution if T 2 = 1, and a projection if T 2 = T . If dim V < and T is an endomorphism of V , show that T
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern