exam_sol - Winter 2011 • Math 67 • Linear Algebra...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Winter 2011 • Math 67 • Linear Algebra Midterm Exam – February 9, 2011 Instructions. You have 50 minutes to complete the 5 problems on the exam. All of your answers should be written in complete English sentences. The symbol V will always denote a vector space over the real numbers R . No books, notes, calculators, talking, texting, etc. is allowed. Problem 1. (a) (3 points) Complete the definition: “The vectors ( v 1 ,...,v n ) are linearly independent if . ..” (b) (2 points) Complete the definition: “The vectors ( v 1 ,...,v n ) form a basis of V if . ..” (c) (5 points) Let ( v 1 ,v 2 ,v 3 ) be a basis of V . Prove that ( v 1 ,v 2 ,v 1 + v 3 ) is also a basis of V . (a) ... a 1 v 1 + a 2 v 2 + ··· + a n v n = 0 implies that a 1 = a 2 = ··· = a n = 0. (b) ...they span V and are linearly independent. (c) Proof. Since v 3 = ( v 1 + v 3 )- v 1 , we have that span { v 1 ,v 2 ,v 1 + v 3 } = span { v 1 ,v 2 ,v 3 ,v 1 + v 3 } = span { v 1 ,v 2 ,v 3 } = V....
View Full Document

This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Winter '07 term at UC Davis.

Page1 / 2

exam_sol - Winter 2011 • Math 67 • Linear Algebra...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online