Section 5 Notes - MATH 294 Chapter 3.2 Subspaces of R n ;...

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Unformatted text preview: MATH 294 Chapter 3.2 Subspaces of R n ; Bases and Linear Independence Ex 3.2.5: Give a geometrical description of all subspaces of R 3 . Answer: Keep in mind that any subspace must contain origin. So subspaces are { ~ } , lines, and planes containing the origin. And dont forget R 3 itself. Ex 3.2.6: Consider two subspaces V and W of R n . a.) Is the intersection U = V T W necessarily a subspace of R n ? Solution: Both V and W are subspaces. Then they contain ~ 0. So, U contains ~ 0 (as U is the intersection of V and W). If ~x and ~ y are in U then both ~x and ~ y are in V and W at the same time. As V and W are subspaces then ~x + ~ y is in V and W then ~x + ~ y is in the intersection of V and W, that is, in U. The same reasoning works for checking k~x U for any ~x U and any constant k R . b.) Is the union U = V S W necessarily a subspace of R n . No, it is not. Because if ~x V and ~ y W then ~x,~ y U = V S W but ~x + ~ y is not necessarily in U. Consider the following example. Let n=3 (so U and W are subspaces of R 3 ) and V= span { (1 , , 0) } and W=span { (0 , 1 , 0) } then U contains all vectors in R 3 having form...
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This note was uploaded on 09/26/2007 for the course MATH 2940 taught by Professor Hui during the Fall '05 term at Cornell University (Engineering School).

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Section 5 Notes - MATH 294 Chapter 3.2 Subspaces of R n ;...

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