Section 5 Notes - MATH 294 Chapter 3.2 Subspaces of Rn...

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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 { 0 } , lines, and planes containing the origin. And don’t forget R 3 itself. Ex 3.2.6: Consider two subspaces V and W of R n . a.) Is the intersection U = V 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 kx U for any x U and any constant k R . b.) Is the union U = V W necessarily a subspace of R n . No, it is not. Because if x V and y W then x, y U = V 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 , 0) } and W=span { (0 , 1 , 0) } then U contains all vectors in R 3 having form ( λ 1 , 0 , 0) or (0
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