Lecture 9: Vector spaces Course Instructor: Dr. Surabhi Tiwari 1) Let V be the set of...

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Lecture 9: Vector spaces Course Instructor: Dr. Surabhi Tiwari 1) Let V be the set of all ordered pairs (x, y), where x, y are real numbers. Let a= (x, y,) and b-(x,, y) be two elements in V. Define the addition as a +b = (*,, y, ) +(x, y; ) =(2x, - 3x,, y, - y.) and the scalar multiplication as a (x,, y,) = (ax,/3,a y, /3). Show that V is not a vector space. Which of the properties are not satisfied? 2) Decide if the following statements are true or false, Provide arguments or a couterexample to support Lecture 9: Vector spaces Course Instructor: Dr. Surabhi Tiwari 1) Let V be the set of all ordered pairs (x, y), where x, y are real numbers. Let a= (x, y,) and b-(x,, y) be two elements in V. Define the addition as a +b = (*,, y, ) +(x, y; ) =(2x, - 3x,, y, - y.) and the scalar multiplication as a (x,, y,) = (ax,/3,a y, /3). Show that V is not a vector space. Which of the properties are not satisfied? 2) Decide if the following statements are true or false, Provide arguments or a couterexample to support