{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

sample_tt1_1_ans

# sample_tt1_1_ans - Math 235 Sample Midterm 1 Answers NOTE...

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

Math 235 Sample Midterm - 1 Answers NOTE : - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Give the definition of an inner product h , i on a vector space V . Solution: h , i : V × V R such that h ~v,~v i > 0 for all ~v 6 = ~ 0 and h ~ 0 , ~ 0 i = 0. h ~v, ~w i = h ~w,~v i h ~v, a~w + b~x i = a h ~v, ~w i + b h ~v, ~x i b) Let B = { ~v 1 , . . . ,~v n } be orthonormal in an inner product space V and let ~v V such that ~v = a 1 ~v 1 + · · · + a n ~v n . Prove that a i = < ~v,~v i > . Solution: Taking the inner product of both sides with ~v i to get < ~v,~v i > = < a 1 ~v 1 + · · · + a n ~v n ,~v i > = a 1 < ~v 1 ,~v i > + · · · + a n < ~v n ,~v i > = a i since B is orthonormal. c) Define what it means for a set B to be orthonormal in an inner product space V . Solution: A set B = { ~v 1 , . . . ,~v n } is orthonormal in V if < ~v i ,~v j > = 0 for all i 6 = j and < ~v i ,~v i > = 1 for 1 i n . d) State the Rank-Nullity Theorem. Solution: Suppose that V is an n -dimensional vector space and that L : V W is a linear mapping into a vector space W . Then rank( L ) + nullity( L ) = dim V = n e) Find the rank and nullity of the linear mapping L : R 3 M 2 × 2 ( R ) defined by L ( x 1 , x 2 , x 3 ) = x 1 x 1 + x 2 x 2 x 1 - x 2 Solution: nullity( L ) = 1 and rank( L ) = 2 2. Let V be an n -dimensional vector space, and let T : V V be defined by T ( ~v ) = λ~v for all ~v in V , where λ R is a constant.

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