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# 603 - Example S S.0 Example S = Rn S = cfw_0 S = cfw_0 S =...

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1 S S . 0 Example: Example: S = R n S = { 0 }, S = { 0 } S = R n . Example: W = { [ w 1 w 2 0 ] T | w 1 w 2 R .} V = { [ 0 0 v 3 ] T | v 3 R .} V = W : (1) for all v V and w W , v w = 0 V W ; (2) since e 1 , e 2 W , all z = [ z 1 z 2 z 3 ] T W must have z 1 = z 2 = 0 by z e 1 = z e 2 = 0 W V . Proposition: For any nonempty subset S of R n , (Span S ) = S . Proof You show it. Corollary: W : a subspace of R n , and B : a basis of W . Then B = W . Example: For W = Span{ u 1 , u 2 }, where u 1 = [ 1 1 1 4 ] T and u 2 =[ 1 1 1 2 ] T , v W if and only if u 1 v = u 2 v = 0 i.e., v = [ x 1 x 2 x 3 x 4 ] T satisfies Property: 1. 0 S for every nonempty subset S of R n . 2. If v , w S for a nonempty subset S of R n , then v + w S and c v S for any scalar c R . Proof First part of 2: ( v + w ) u = v u + w u = 0 + 0 = 0, u S .

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2 is a basis for W . Proof v (Row A ) w v = 0 for all w Span{rows of A } A v = 0 . Also, (Col A ) = (Row A T ) = Null A T . For any real matrix A , (Row A ) = Null A , or (Col A ) = Null A T . * For a complex matrix A and orthogonal complement defined by the complex dot product, (Row A ) = Null A and (Col A ) = Null
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603 - Example S S.0 Example S = Rn S = cfw_0 S = cfw_0 S =...

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