hw2sol - Winter 2011 Math 67 Linear Algebra Homework 2...

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Winter 2011 Math 67 Linear Algebra Homework 2 Problems 1.10, 1.13, 1.14. from Axler. Problem 1.10. What is U + U ? The answer is U . The reason is that U + U consists of linear combinations of elements from U . But U is closed under taking linear combinations, since it is a subspace. Hence U + U U . Next we see that U U + U . This is because we can write any element u U as 0 + u U + U . We conclude that U U + U U = U = U + U. Problem 1.13. The question is whether U 1 + W = U 2 + W implies that U 1 = U 2 . This is false, which means we need to find a single counterexample. Here’s just one counterexample; there are many. Take three different lines passing through the origin in R 2 . Each of these lines determines a subspace in R 2 , call them U 1 ,U 2 ,W . We know that U 1 + W = U 2 + W = R 2 . However U 1 6 = U 2 , as required. Problem 1.14. Let W be the span of the monomials { z n : n = 0 , 1 , 3 , 4 , 6 , 7 , 8 , 9 ,... } , i.e., W is the span of z n
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hw2sol - Winter 2011 Math 67 Linear Algebra Homework 2...

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