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Unformatted text preview: Partial Solution Set, Leon 3.2 3.2.1 Determine whether the following are subspaces of R 2 . (b) S = { ( x 1 ,x 2 ) T  x 1 x 2 = 0 } No, this is not a subspace. Every element of S has at least one component equal to 0. The set is closed under scalar multiplication, but not under addition. For example, both (1 , 0) T and (0 , 1) T are elements of S , but their sum is not. (c) S = { ( x 1 ,x 2 ) T  x 1 = 3 x 2 } . Yes, this is a subspace. To see this, let x = (3 x,x ) T and y = (3 u,u ) T . Both are elements of S . For any scalar , x = (3 x,x ) S . Also x + u = (3 x + 3 u,x + u ) T = (3( x + u ) ,x + u ) T S . 3.2.3 Determine whether the following are subspaces of R 2 2 . (a) The set of all 2 2 diagonal matrices is a subspace of R 2 2 , since a scalar multiple of a diagonal matrix is diagonal and the sum of two diagonal matrices is diagonal. (c) The set of all 2 2 matrices such that a 12 = 1 is not a subspace of R 2 2 , since it is closed under neither scalar multiplication nor addition.closed under neither scalar multiplication nor addition....
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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, Scalar

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