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**Unformatted text preview: **Test 1 Solutions Throughout, F is a field. 1. Suppose V is a vector space over F . Prove that for all a F , a ~ 0 = ~ where ~ is the zero vector of V . (This is a theorem from the course.) For all a F , a ~ 0 = a ( ~ 0 + ~ 0) = a ~ 0 + a ~ 0. Then subtract a ~ 0 from both sides to get ~ 0 = a ~ 0. 2. (a) Define the term basis . A basis is a subset of a vector space V which is linearly independent and which spans V . (b) Suppose V is a vector space, S a finite spanning set of V , and L a linearly in- dependent subset of S . Prove that V has a basis B with L B S . (This is a theorem from the course. Do not deduce this using another theorem which asserts the existence of a basis.) Let S = { C S | L C and C is linearly independent } Since L S , the set S is non-empty. Since S is finite, it has only a finite number of subsets, so S is a finite set. Therefore, it contains a maximal element B . Then L B S and B is linearly independent. Suppose there exists v S- span( B ). Then L B B { v } S . Since v 6 span( B ), B { v } is linearly independent, so B { v } S (and v 6 B since v 6 span( B )). This contradicts the maximality of B . Thus S span( B ), which implies V = span( S ) span(span( B )) = span( B ) which implies B spans as well, so it is a basis. 3. Suppose V , W , and X are vector spaces, T : V W and U : W X linear transformations. (a) Define null space (i.e. N ( T ) ) and nullity (i.e. nullity ( T ) ). The null space is N ( T ) = { v V | T ( v ) = ~ } . and nullity( T ) = dim( N ( T )). (b) Prove that N ( UT ) = { v V | T ( v ) N ( U ) } ....

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