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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 5: Nullspace. Column Space. Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay Consider the subspace f x : Ax = g . This is called the nullspace of the matrix A . By the column space of A , we mean the vector space spanned by the columns of A . We wish to relate the dimension of the nullspace with the dimension of the space spanned by the vectors which form the columns of A . How do we go about doing this? The column perspective is best understood from the following equivalence. Ax = b , A (1) x 1 + A (2) x 2 + : : : + A ( n ) x n = b Note that this says that Ax = b has a solution i b is in the column space of A . Here, A ( i ) is the i th column of A and x i is the i th component of vector x . We need to nd all x such that A (1) x 1 + A (2) x 2 + : : : + A ( n ) x n = Our objective is to prove that the dimension of f x : Ax = g is n k where k is the dimension of the column space of A . How do we begin such a proof? What should the structure of such a proof look....
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- Summer '09
- Computer Science