01aut-m2sols

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/12/2010 for the course MATH 51 at Stanford.

Page1 / 12

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online