Question

v = eue" for every 6. Let V be the set of positive real numbers. Define the operations u u and v in V, and cou ecu for every real number c and every u in V. Which of the following statements are TRUE? (i) V is closed under . () V is closed under O. (iii) u田v=v田ufor every u and v in V. (iv) 1Ou- u for every u in V. A. (i) and (iii) but not (ii) B. ) and (iv) C. (i and (ii) but not (iii D. i and (iv) E. (i) and (ii) and (iii)

Linear Algebra Subspaces.

Correct answer is E - please help explain why. I do not understand the concept. Thanks!

Answer 1

Note: In order to show that a vector space U is closed under an operation *, we have to show that for u, v elementof U we have u * v elementof V. So, here first i will show (i) is true. We have to show V is closed under So, for u, v elementof V we have to show u v elementof V. Given, u v = e^u middot e^v. We know V is the set of positive real's. So, we need to show e^u middot e^v is also a positive real (then obviously it belong to V). \r\n n We know that for any real number x, e^x is positive. Also, e^x is a real number. So, we have e^u and e^v are real numbers. We know if a and b are real's then ab is also real - So, e^u e^v is a real number. Since e^u and e^v are positive, e^u e^v