College of Science First Semester 20162017 Maths 21 Linear A

College of Science First Semester 2016/2017 Maths 21 Linear Algebra for Mechanical Engineering Date 29 November 2016 Dr Thuraya Juma sec. I) Max Marks 10 Question 1: 12 markal Let N-a 2,A4 u. ie. set of natural numbers with the standard addition and scalar multiplication. Is a real vector space. Justify your answer. Question 2: 12 marksl det A o with the standard scalar multiplication and followi Show that the set v A- addition A+ B AB is not a vector space. TJA

Solution

The set N as described here does not contain the additive identity 0 and nor the additive inverse of any element. Hence N is not a vector space. If det(A) 0, then det(-A). Thus if A V, then –AV. However, since det[ A +(-A) ] = 0 , therefore A + (-A) does not belong to V. Hence V is not closed under addition so that V is not a vector space.