rndecomp - The Range-Nullspace Decomposition of C n Math 422

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Unformatted text preview: The Range-Nullspace Decomposition of C n Math 422 De&nition 1 Let A be an n & n matrix. The range (or column space) of A ; denoted by R ( A ) ; is the subspace of C n spanned by the columns of A: The nullspace of A , denoted by N ( A ) ; is the solution space of the linear system Ax = 0 : De&nition 2 Let U and V be subspaces of a vector space W such that U \ V = 0 : The direct sum of U and V , denoted by U ¡ V , is the subspace f u + v j u 2 U and v 2 V g : When U ¡ V = W; U and V are called complementary subspaces and U ¡ V is called a direct sum decomposition of W . Exercise 3 Prove that if w 2 U ¡ V; there exist unique elements u 2 U and v 2 V such that w = u + v . Example 4 The subspaces U = f ( x;y; 0) j x;y 2 R g and V = f (0 ; ;z ) j z 2 R g are complementary sub- spaces of R 3 ; U ¡ V is a direct sum decomposition of R 3 : An element w = ( x;y;z ) 2 R 3 is the unique sum w = u + v; where u = ( x;y; 0) 2 U and v = (0 ; ;z ) 2 V: The range and nullspace of an n & n complex matrix A may or may not intersect trivially. The Dimension Theorem tells us that dim R ( A )+dim N ( A ) = n ; so if A is non-singular, then N ( A ) = 0 and R ( A ) \ N ( A ) = : But if A is singular, it is possible that R ( A ) \ N ( A ) 6 = 0 as the next example shows. Example 5 Let A = & 1 ¡ , then R ( A ) = N ( A ) = ¢£ 1 ¤¥ : However, note that A 2 = 0 ; in which case R ¦ A 2 § = 0 ; N ¦ A 2 § = C 2 and R ¦ A 2 § \ N ¦ A 2 § = 0 : Theorem 6 (The Range-Nullspace Decomposition of C n ) If A is an n & n complex matrix, there exists a...
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rndecomp - The Range-Nullspace Decomposition of C n Math 422

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