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Unformatted text preview: Partial Solution Set, Leon Â§ 3.6 3.6.1b We want bases for the row space, the column space, and the nullspace of A =  3 1 3 4 1 2 1 2 3 8 4 2 . Elimination transforms A to U =  3 1 3 4 0 7 0 2 0 0 1 . We have one free variable and three nonzero pivots. A basis for the row space of A can be either all rows of A or all rows of U . A basis for the column space of A consists of the first three columns of A . A basis for the nullspace of A is B = { (10 , 2 , , 7) T } . 3.6.2c What is the dimension of the subspace of R 3 spanned by (1 , 1 , 2) T , ( 2 , 2 , 4) T , (3 , 2 , 5) T , (2 , 1 , 3) T ? Solution : The solution is simply the rank of A = 1 2 3 2 1 2 2 1 2 4 5 3 . We use Gaussian elimination to obtain an equivalent matrix B = 1 2 3 2 0 1 1 0 0 0 , from which it is apparent that rank( A ) = 2. Thus the dimension of the subspace in question is 2. 3.6.3 Given A = 1 2 2 3 1 4 2 4 5 5 4 9 3 6 7 8 5 9 , (a) Compute the reduced row echelon form U of A . Which columns of U correspond to the free variables? Write each of these vectors as a linear combination of the column vectors corresponding to the lead variables. (b) Which columns of A correspond to the lead variables of U ? These column vectors constitute a basis for CS ( A ). Write each of the remaining column vectors of A as a linear combination of these basis vectors. Solution : (a) The reduced row echelon form of A is U = 1 2 0 5 3 0 0 0 1 1 2 0 0 0 0 0 1 . The columns corresponding to free variables are u 2 , u 4 , and u 5 . Because of the simplicity of the column vectors corresponding to the lead variables, it is easy to see that u 2 = 2 u 1 , u 4 = 5 u 1 u 3 , and u 5 = 3 u 1 + 2 u 3 . (b) The column vectors of A corresponding to the lead variables of U are a 1 , a 3 , and a 6 . What is perhaps not immediately obvious is that the dependencies among the columns of A are precisely the same as those among the columns of...
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This note was uploaded on 07/17/2011 for the course MATH 311 taught by Professor Anshelvich during the Spring '08 term at Texas A&M.
 Spring '08
 Anshelvich
 Linear Algebra, Algebra

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