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2. Criticize the following argument: By Exercise 1.1.13, for any vector v, we have Ov 0. So the first criterion for subspaces is, in fact, a consequence of the second criterion and could therefore be omitted. 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 , CVk s * 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". Define UnV={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 V Prove that U + V is a subspace of R". Give two examples. VkER" and let v e R". Prove that 7. Let v1, ... , Snan(v Snan(v (x Sna Y

2. Criticize the following argument: By Exercise 1.1.13, for any vector v, we have Ov 0. So the first criterion for subspaces is, in fact, a consequence of the second criterion and could therefore be omitted. 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 , CVk s * 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". Define UnV={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 V Prove that U + V is a subspace of R". Give two examples. VkER" and let v e R". Prove that 7. Let v1, ... , Snan(v Snan(v (x Sna Y

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 30EQ
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For problem 6. Please help me the following:

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

- Please help me solve 6b.

Thank you very much!

2. Criticize the following argument: By Exercise 1.1.13, for any vector v, we have Ov 0. So the first criterion for subspaces is, in fact, a consequence of the second criterion and could therefore be omitted. 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 , CVk s * 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". Define UnV={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 V Prove that U + V is a subspace of R". Give two examples. VkER" and let v e R". Prove that 7. Let v1, ... , Snan(v Snan(v (x Sna Y
Transcribed Image Text:2. Criticize the following argument: By Exercise 1.1.13, for any vector v, we have Ov 0. So the first criterion for subspaces is, in fact, a consequence of the second criterion and could therefore be omitted. 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 , CVk s * 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". Define UnV={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 V Prove that U + V is a subspace of R". Give two examples. VkER" and let v e R". Prove that 7. Let v1, ... , Snan(v Snan(v (x Sna Y
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