MATH311Midterm2Practice

# MATH311Midterm2Practice - Math 311.503/505 Practice for...

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

Math 311.503/505 - Practice for Midterm 2 1. Answer TRUE or FALSE. – i) Let k · k be a norm in R n . Then for every x, y R n one has that k x - y k 2 + k x + y k 2 = 2 k x k 2 + 2 k x k 2 . – ii) Let A be a 5 × 3 matrix with rank( A ) = 2. Then dim N ( A ) = 1. – iii) Let S be a subspace of R n . Then the only element in the intersection of S and S is the zero vector. – iv) Let A be an m × n matrix. Then the number of linearly independent columns is equal to the number of linearly independent rows. – v) Let A, B, C be n × n matrices. Then tr( ABC ) = tr( CAB ). – vi) The functions 1, x 2 are orthogonal in the space C ([ - 1 , 1]) with inner prod- uct h f, g i := R 1 - 1 f ( x ) g ( x ) dx . – viii) If x, y non-zero vectors in R 2 with h x, y i = 0 have the property that span { x 1 , x 2 } = R 2 . 2. Determine whether the following are linear transformations: 1. L : R n × n R n × n , L ( A ) = L ( A T ) , 2. L : C ([0 , 1]) R 2 , L ( f ) := R 1 0 f ( x ) dx, f (0)+ f (1) 2 T , 3. L : R 2 R 3 , L ( f ) := ( x 1 , x 1 + x 2 , x 1 x 2 ) T .

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