01aut-m2sols

# 01aut-m2sols - MATH 51 MIDTERM 2 SOLUTIONS (AUTUMN 2001) 1....

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MATH 51 MIDTERM 2 SOLUTIONS (AUTUMN 2001) 1. Let A = 1 2 0 3 1 2 4 - 1 5 4 3 6 - 1 8 5 4 8 - 1 12 8 . Then rref( A ) = 1 2 0 0 - 5 0 0 1 0 - 4 0 0 0 1 2 0 0 0 0 0 . (You do not need to verify this.) (a) (3 points) Find a basis for C ( A ). Solution. The ﬁrst, third, and fourth columns of rref( A ) have pivots, so 1 2 3 4 , 0 - 1 - 1 - 1 , 3 5 8 12 is a basis for C ( A ). (b) (4 points) Express each column of A which is not in your basis for C ( A ) as a linear combination of your basis columns. Solution. Denote the columns of A by v 1 through v 5 . Then v 1 , v 3 and v 4 are the basis vectors from part (a). By inspection v 2 = 2 v 1 . Looking at the columns w 1 through w 5 of rref( A ), it is clear that w 5 = - 5 w 1 - 4 w 3 + 2 w 4 . Therefore, since the columns of A and rref( A ) share the same relations, it follows that v 5 = - 5 v 1 - 4 v 3 + 2 v 4 . (c) (3 points) What is the maximum number of linearly independent vectors which can be found in N ( A )? Solution. 2. The rank of A is 3, so the nullity of A is 2 by the Rank-Nullity Theorem. 1

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2. (10 points) Answer each question True or False. No explanation is necessary. (a) If V is a linear subspace of R 5 and V 6 = R 5 then any set of 5 vectors in V is linearly dependent. Solution. True. (Any set of 5 linearly independent vectors in R 5 must span R 5 .) (b) If A is a 4 × 7 and if the dimension of N ( A ) is 3 then for any b in R 4 the system A x = b has inﬁnitely many solutions. Solution. True. (By the Rank-Nullity Theorem, rank( A ) = 4, which implies that C ( A ) = R 4 . Therefore, A x = b has at least one solution for every b R 4 . Since the null space of A is nontrivial, the set of solutions for each b is inﬁnite.) (c) If T : R 5 R 3 is an onto linear transformation and if { v 1 , v 2 , v 3 } is a linearly independent set of vectors in R 5 , then { T ( v 1 ) , T ( v 2 ) , T ( v 3 ) } spans R 3 . Solution.
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## This note was uploaded on 01/12/2010 for the course MATH 51 at Stanford.

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01aut-m2sols - MATH 51 MIDTERM 2 SOLUTIONS (AUTUMN 2001) 1....

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