# sol_310f04_2 - Problem Points Scores 12 6 12 3 10 4 10 5 12...

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Problem 1 2 3 4 5 Bonus: Total: Points 6 12 10 10 12 10 50+10 Scores Mat 310 – Linear Algebra – Fall 2004 Name: Id. #: Lecture #: Test 2 (November 05 / 60 minutes) There are 5 problems worth 50 points total and a bonus problem worth up to 10 points. Show all work. Always indicate carefully what you are doing in each step; otherwise it may not be possible to give you appropriate partial credit. 1. [6 points] Let W 1 and W 2 be linear subspaces of a vector space V such that W 1 + W 2 = V and W 1 W 2 = { 0 } . Prove that for each vector α V there are unique vectors α 1 W 1 and α 2 W 2 such that α = α 1 + α 2 . Solution: Since W 1 + W 2 = V , every vector α V can be represented as α = α 1 + α 2 for some α 1 W 1 and α 2 W 2 . We must then show that this representation is unique. Suppose that there are two other vectors β 1 W 1 and β 2 W 2 such that α = β 1 + β 2 . Then α - α = 0 = ( α 1 + α 2 ) - ( β 1 + β 2 ) so α 1 - β 1 = β 2 - α 2 . Call the above vector γ . Then since α 1 and β 1 are both in W 1 , their diﬀerence γ must also be in W 1 . Similarly, γ must be in W 2 , and so γ W 1 W 2 . Therefore γ = 0 , that is, α 1 = β 1 and α 2 = β 2 .

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2. [12 points] Consider the vectors in R 4 deﬁned by α 1 = ( - 1 , 0 , 1 , 2) , α 2 = (3 , 4
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sol_310f04_2 - Problem Points Scores 12 6 12 3 10 4 10 5 12...

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