{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

exam2math3042009solns

# exam2math3042009solns - Exam 2 Math 304 2009 1 Dene the...

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

Exam 2 Math 304, 2009 1. Define the following terms, or answer true-false, as appropriate: (The carefully worded definitions can be looked up in your book. I’ll provide one such, and for the rest, I’ll limit the definition to a sketch that includes the essential core information. (a) a linear transformation from a vector space U to a vector space V . A linear transformation L from U to V is a function L from U to V such that for all u 1 and u 2 in U , and all scalars α , L ( u 1 + u 2 ) = Lu 1 + Lu 2 , and L ( αu 1 ) = αL ( u 1 ). Short version: L ( αu 1 + βu 2 ) = αLu 1 + βLu 2 . (b) the kernel of a linear operator on U . { u : Lu = 0 } . (c) an inner product on a vector space U , when the field is R . ( u, v ) = ( v, u ) , ( u, v + w ) = ( u, v ) + ( u, w ) , and ( αu, v ) = α ( u, v ) . (d) an orthogonal matrix . Q T = Q - 1 , OR, Q is a square matrix whose columns form an orthonormal set. (e) the orthogonal complement of a vector subspace V of an inner product space U . { u U such that u v v V } . (f) an orthonormal set of vectors in an inner product space. The vectors are mutually orthogonal and all have length 1. (g) a unit vector in an inner product space. A vector v so that ( v, v ) = 1. Also, but not as good a definition because it ducks the question of how length is measured, a vector v with bardbl v bardbl = 1. (h) The product of two orthogonal matrices is again an orthogonal ma- trix. True. Reasons weren’t required, but here’s the reason: if P and Q are orthogonal n × n matrices, then Q T Q = I and P T P = I .

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 ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern