1 Let V be the set of all vectors of the form for all real

1. Let V be the set of all vectors of the form <a, 3a+7b, b>, for all real numbers a and b. Is V a vector space under the usual operations of addition and scalar multiplication? Explain.

Solution

V = < a , 3a +7b , b >

a) Property of vector addition : <a1 , 3a1 +7b1 , b1 > + < a2 , 3a2 + 7b2 , b2 >

= < a1 +a2 , 3(a1 +a2) +7(b1 +b2) , b1 + b2 >

= < a1, 3a1 +3b1 , b1 > + < a2, 3a2 +3b2 , b2 >

b) Property of scalar multiplication

= c<a ,3a +7b , b >

= <ac , 3ac +7bc , bc >

Since V follows the thabove two properties it is a vector space