C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes4-2

C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes4-2 -...

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SECTION 4.2 IMPORTANT EXAMPLES OF SUBSPACES A matrix A generates two subspaces. 1. The null space of A is the set of all solutions of A x = 0 , written as Nul A . 2. The column space of A is the span of the columns of A , written as Col A . Are these things really subspaces, and if so, of what exactly? Nul A Col A
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Closely related to Nul A and Col A are two subspaces generated by a linear transforma- tion. We can now talk about a linear transformation T from a vector space V to a vector space W . What does this mean? The kernel or null space of a linear transformation T is the set of all vectors u for which T ( u ) = 0 . The range of a linear transformation T is the set of all images of T , that is, the set of
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Unformatted text preview: all T ( x )s. EXAMPLES (1) Find an explicit description of Nul 1 3 0-2 0 0 0 1-4 0 0 0 0 0 1 (2) Why is p q r : 3 p + 2 q = 5 r a subspace and what kind of subspace is it, exactly? (3) Why is p q r : 3 p + 2 q = 5 r + 1 not a subspace? (4) For A = 1 0-2 0 0 1-4 0 0 0 0 1 , where do Nul A and Col A live? Find nonzero vectors in each. HOMEWORK: SECTION 4.2...
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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes4-2 -...

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