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sol32 - Partial Solution Set Leon 3.2 3.2.1 Determine...

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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 α , α x = (3 αx, αx ) 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.
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