midterm-2

# midterm-2 - Spring 2010 CS530 Analysis of Algorithms...

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

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.

Unformatted text preview: Spring 2010 CS530 Analysis of Algorithms Midterm exam 2 M IDTERM EXAM 2 Only a single hand-written crib sheet can be used, no books or notes. Even if I ask for just a yes / no answer, you must always give a proof. You may get some points even if you write I dont know, but if you write something that is wrong, you may get less. (It is not possible to pass just writing I dont know everywhere....) Problem 1. (10pts) Given n n matrices A and B , prove rank ( A + B ) rank ( A ) + rank ( B ) . Solution. The rank of a matrix is the dimension of its column space. Let e 1 ,..., e r be a basis of the column space of A , and f 1 ,..., f s be a basis of the column space of B . Then every column of A is a linear combination of the vectors e i : say, a 1 = 1 e 1 + + r e r . Similarly, every column of B is a linear combination of the vectors f j : say, b 1 = 1 f 1 + + s f s . Then every column of A + B is a linear combination of the vectors e i , f j : say, a 1 + b 1 = 1 e 1 + + r e r + 1 f 1 + + s f s . Problem 2. (10pts) Consider the following linear program. maximize 2 x 1 + 5 x 2 + 6 x 3 + x 4 + 3 x 5 subject to 2 x 1- 3 x 2 + x 3 + 4 x 4 + 5 x 5 10,- 3 x 1 + 4 x 2 + 5 x 3 + x 4 + 9 x 5 5, x...
View Full Document

## This document was uploaded on 10/05/2010.

### Page1 / 3

midterm-2 - Spring 2010 CS530 Analysis of Algorithms...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online