MAT224H1 Midterm: MAT224 MIDTERM SELF GENERATED SOLUTION 2009 PART2.pdf
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Let v be the set of ordered pairs (x, y) of real numbers with the operations of vector addition and scalar multiplication given by (x, y) + (x(cid:48), y(cid:48)) = (x + x(cid:48), yy(cid:48)) c(x, y) = (cx, y) Solution: checking the list of properties that a vector space must satisfy, via property. 3 we nd that the additive identity is (0, 1) since for all (a, b) v we have (a, b) + (0, 1) = (a + 0, b 1) = (a, b). Then we check property 4 for all (a, b) v and we nd that if b (cid:54)= 0, ( a, 1/b) is an inverse since (a, b) + ( a, 1/b) = (a a, b/b) = (0, 1). Let v and w be vector spaces over a eld f . Show that the range (image) of t is a subspace of w .
