math54-fall07-mt1-Gu-soln

math54-fall07-mt1-Gu-soln - Prof Ming Gu 861 Evans tel...

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Unformatted text preview: Prof. Ming Gu, 861 Evans, tel: 2-3145 Email: [email protected] http://www.math.berkeley.edu/ ∼ mgu/MA54 Math54 Midterm I Solutions This is a closed everything exam, except a standard one-page cheat sheet (on one- side only). You need to justify every one of your answers. Completely correct answers given without justification will receive little credit. Problems are not necessarily ordered according to difficulties. You need not simplify your answers unless you are specifically asked to do so. Problem Maximum Score Your Score 1 5 2 19 3 19 4 19 5 19 6 19 Total 100 1. (5 Points) Write your personal information below. Your Name: Your GSI: Your SID: Math 54 Midterm I Solutions 2 2. (19 Points) Show that you need at least m vectors to span a linear space of dimension m . Proof: Let v 1 , ··· , v k be a set of spanning vectors. Remove all the redundant vectors from v 1 , ··· , v k . The resulting set of vectors must be linearly independent and still span the linear space, and hence must be a basis for the linear space. By definition, the number of vectors in this basis is the dimension. It follows that m ≤ k . Math 54 Midterm I Solutions...
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math54-fall07-mt1-Gu-soln - Prof Ming Gu 861 Evans tel...

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