Macroeconomics Exam Review 18

Macroeconomics Exam Review 18 - Similarly x =(0 2 3 =(0 2 3...

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Unformatted text preview: Similarly x = (0 , 2 , 3 ,..., ) = (0 , 2 , 3 ,..., ) ∈ Therefore is a subspace of ℜ . Generalizing, any set of vectors with one or more coordinates identically zero is a subspace of ℜ . 4. We will meet some more complicated subspaces in Chapter 2. 1.127 No, − x / ∈ ℜ + if x ∈ ℜ + unless x = 0. ℜ + is an example of a cone (Section 1.4.5). 1.128 lin is a subspace Let x , y be two elements in lin . x is a linear combination of elements of , that is x = 1 1 + 2 2 + ... Similarly y = 1 1 + 2 2 + ... and x + y = ( 1 + 1 ) 1 + ( 2 + 2 ) 2 + ⋅⋅⋅ + ( + ) ∈ lin and x = 1 1 + 2 2 + ⋅⋅⋅ + ∈ lin This shows that lin is closed under addition and scalar multiplication and hence...
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