hw3 - A T ). (a) Show that b R and b N are unique. (b) Show...

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Math 171A: Linear Programming Instructor: Philip E. Gill Winter Quarter 2011 Homework Assignment #3 Due Friday January 28, 2011 Remember that the frst midterm exam will be held in class on Wednesday, January 26. Starred exercises require the use oF Matlab . Exercise 3.1. Let A denote an m × n matrix. (a) If A has rank m , what does this imply about the relative sizes of m and n ? (b) Answer part (a) , this time assuming that A has rank n . (c) Show that when A has rank n , any solution (if it exists) of Ax = b is unique. Exercise 3.2. * (a) De±ne range( A ), null( A T ), range( A T ), null( A ) and rank( A ) for a general matrix A . (b) Consider the matrix A = 3 2 1 4 1 2 2 3 0 5 0 3 . (i) Use the Matlab command rank to ±nd rank( A ). (ii) Find the dimensions of the subspaces range( A ), null( A T ), range( A T ) and null( A ). (iii) Use the Matlab command null to ±nd a basis for null( A T ). Exercise 3.3. For a given nonzero matrix A and nonzero vector b , assume that b may be written as b = b R + b N , where b R range( A ) and b N null(
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Unformatted text preview: A T ). (a) Show that b R and b N are unique. (b) Show that b T R b N = 0. (c) If b R and b N are both nonzero, show that they are linearly independent. Exercise 3.4. * Consider the matrix A and vector y given by A = 7-7-7 2 4 6 7 7 7 14 6 2-2 5 3 1 4 9 6 3 9 5 2-1 1 and y = -7-3 1-1 . (a) Find the rank of A . Use the Matlab command null to nd a basis for null( A ). (b) Find the unique vectors y R range( A T ) and y N null( A ) such that y = y R + y N . Exercise 3.5. Let A be an m n matrix with rank m . Use the unique decomposition of a vector into the sum of its null- and range-space portions to show that the system Ax = b is compatible for every b R m ....
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