# 2. Criticize the following argument: By Exercise 1.1.13, for any vector v, we have Ov 0. So thefirst criterion for subspaces is, in fact, a consequence of the second criterion and could therefore beomitted.3. Suppose x, Vi, . .,orthogonal to any linear combination civi +C2V2 +.+Vk E R" and x is orthogonal to each of the vectors v1,..Vh. Prove that x is,CVks *4. Prove Proposition 3.2.5 Given vectors v1, ..V) is the smallest subspace con-Vk E R", prove that V = Span (v1, ..taining them all. That is, prove that if W C R" is a subspace and vi, ..., Vk E W, then VC W>.6. (a) Let U and V be subspaces of R". DefineUnV={x e R" : xe U and x e V}.Prove that U nVis a subspace of R". Give two examples.(b) Is U UV= {x e R" : x eU or x e V}a subspace of R"? Give a proof or counterexample.(c) Let U and V be subspaces of R". Define{x eR" : x = u + v for some u e U and v e V}.U VProve that U + V is a subspace of R". Give two examples.VkER" and let v e R". Prove that7. Let v1, ... ,Snan(vSnan(v(xSnaY

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-For 6a and 6c. I have done the proof. Please only give me the two examples for each.

Thank you very much!

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Step 1

To give examples for 6(a), 6(c) and counterexamples for 6 (b)

Step 2

6(a) Example 1: Here the intersection of the two subspaces is the 0 vectors subspace.

Step 3

6(a) Example 2: Here , the intersection of the subspaces ...

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