bergman_notes

# bergman_notes - Math 110 Midterm Exam Professor K A Ribet...

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2. (9 points) Let V be a vector space over a field F . Suppose that v 1 , . . . , v n are ele- ments of V and that w 1 , . . . , w n , w n +1 lie in the span of { v 1 , . . . , v n } . Show that the set { w 1 , · · · , w n +1 } is linearly dependent. Let W be the span of { v 1 , . . . , v n } . Then W is generated by n elements, so its dimension d is at most n (for example, by Theorem 1.9 on page 42). If the vectors w i were linearly independent, the set { w 1 , · · · , w n +1 } could be extended to a basis of W . This is impossible because all bases of W have d elements. 3. (10 points) Let W 1 and W 2 be subspaces of a finite-dimensional F -vector space V . Recall that W 1 × W 2 denotes the set of pairs ( w 1 , w 2 ) with w 1 W 1 , w 2 W 2 . This product comes equipped with a natural addition and scalar multiplication: ( w 1 , w 2 ) + ( w 1 , w 2 ) := ( w 1 + w 1 , w 2 + w 2 ) , a ( w 1 , w 2 ) := ( aw 1 , aw 2 ) .
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