pset4-s09-soln

# Pset4-s09-soln - 18.06 Problem Set 4 Solution Due Wednesday 11 March 2009 at 4 pm in 2-106 Total 175 points Problem 1 A is an m n matrix of rank r

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18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2-106. Total: 175 points. Problem 1: A is an m × n matrix of rank r . Suppose there are right-hand-sides ~ b for which A~x = ~ b has no solution. (a) What are all the inequalities ( < or ) that must be true between m , n , and r ? (b) A T ~ y = ~ 0 has solutions other than ~ y = ~ 0. Why must this be true? Solution (15 points = 10+5) (a) First of all, the rank r of a matrix is the number of column (row) pivots, it must be less than equal to m and n . If the matrix were of full row rank, i.e., r = m , it would imply that A~x = ~ b always has a solution; we know that this is not the case, and hence r 6 = m . To sum up, the inequalities among m,n,r are r n,r < m . (b) Since A T is an n × m matrix, the null space N ( A T ) has dimension m - r , which is positive by (a). Hence, A T ~ y = ~ 0 has solutions other than ~ y = ~ 0. Problem 2: A is an m × n matrix of rank r . Which of the four fundamental subspaces are the same for: (a) A and ± A A ² (b) ± A A ² and ± A A A A ² Explain why all three matrices A , ± A A ² , and ± A A A A ² must have the same rank r . Solution (20 points = 10+10) (a) Note that if we do invertible row operations on the matrix ± A A ² , we may kill the bottom A and get ± A 0 ² . 1

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Nullspace: Since the nullspace is invariant under row operations, the two matri- ces have the same nullspace. Column space: Since the two matrices do not have the same number of rows, their column space must not be the same. Row space: Since the row space is also invariant under row operations, the two matrices have the same row space. Left nullspace: The transpose of ± A A ² is ( A T A T ) . Hence, the left nullspace is a subspace of R 2 m . However, the left nullspace is a subspace of R m . They cannot be the same. (b) Using invertible column operations, we can turn ± A A A A ² into ± A 0 A 0 ² . The nullspace and the row space of ± A A A A ² are subspaces of R 2 n , whereas the nullspace and the row space of ± A A ² are subspaces of R n . They cannot be the same. Since the column space and the left nullspace are invariant under column oper- ations, the two matrices have the same column space and left nullspace. REMARK: For an m × n matrix of rank r , we have Fundamental space subspace of dimension Nullspace R n n - r Column space R m r Row space R n r Left nullspace R m m - r Problem 3: Find a basis for each of the four subspaces for A = 0 1 2 3 4 0 1 2 4 6 0 0 0 1 2 Solution (25 points = 5(echelon form)+5+5+5+5) We ﬁrst write A as in row-reduced echelon form. 0 1 2 3 4 0 1 2 4 6 0 0 0 1 2 ; 0 1 2 3 4 0 0 0 1 2 0 0 0 1 2 ; 0 1 2 3 4 0 0 0 1 2 0 0 0 0 0 ; 0 1 2 0 - 2 0 0 0 1 2 0 0 0 0 0 2
The second and the fourth variables are the pivots. The ﬁrst, third, and the ﬁfth variables are free variables. The row operation matrix

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## This note was uploaded on 05/06/2010 for the course 18 18.06 taught by Professor Strang during the Spring '08 term at MIT.

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Pset4-s09-soln - 18.06 Problem Set 4 Solution Due Wednesday 11 March 2009 at 4 pm in 2-106 Total 175 points Problem 1 A is an m n matrix of rank r

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