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If v1, v2, v6 are six vectors in R4, which one of the following statements is False?
- a)If {v1 v3, v5, v6} spans R4, then it forms a basis for R4.
- b)These vectors are not linearly independent.
- c)It is not necessary that these vectors span R4.
- d)Any four of these vectors form a basis for R4.
Correct answer is option 'D'. Can you explain this answer?
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If v1,v2, v6 are six vectors in R4, which one of the following stateme...
Explanation:
To determine which statement is false, we need to analyze each statement.
a) If {v1, v3, v5, v6} spans R4, then it forms a basis for R4.
This statement is true. If a set of vectors spans a vector space, and the vectors are linearly independent, then the set forms a basis for that vector space. However, the statement in option a) only mentions {v1, v3, v5, v6}, not {v1, v3, v5, v6}. So, we cannot determine its validity based on the information provided.
b) These vectors are not linearly independent.
This statement is true. If the vectors v1, v2, and v6 are in R4, and they are given to be vectors in R4, then they are not linearly independent. This is because at least one vector can be written as a linear combination of the other vectors in the set.
c) It is not necessary that these vectors span R4.
This statement is true. Just because the vectors v1, v2, and v6 are in R4, it does not mean that they span R4. In order for a set of vectors to span a vector space, they need to be able to generate all possible vectors in that space. The given statement does not provide any information about whether these vectors span R4 or not.
d) Any four of these vectors form a basis for R4.
This statement is false. In order for a set of vectors to form a basis for R4, they need to be linearly independent and span R4. However, we do not have enough information to determine whether any four of these vectors form a basis for R4. Additionally, the statement in option d) does not mention any specific four vectors, so we cannot determine its validity based on the information provided.
Conclusion:
Based on the analysis of each statement, the false statement is option d) "Any four of these vectors form a basis for R4."
To determine which statement is false, we need to analyze each statement.
a) If {v1, v3, v5, v6} spans R4, then it forms a basis for R4.
This statement is true. If a set of vectors spans a vector space, and the vectors are linearly independent, then the set forms a basis for that vector space. However, the statement in option a) only mentions {v1, v3, v5, v6}, not {v1, v3, v5, v6}. So, we cannot determine its validity based on the information provided.
b) These vectors are not linearly independent.
This statement is true. If the vectors v1, v2, and v6 are in R4, and they are given to be vectors in R4, then they are not linearly independent. This is because at least one vector can be written as a linear combination of the other vectors in the set.
c) It is not necessary that these vectors span R4.
This statement is true. Just because the vectors v1, v2, and v6 are in R4, it does not mean that they span R4. In order for a set of vectors to span a vector space, they need to be able to generate all possible vectors in that space. The given statement does not provide any information about whether these vectors span R4 or not.
d) Any four of these vectors form a basis for R4.
This statement is false. In order for a set of vectors to form a basis for R4, they need to be linearly independent and span R4. However, we do not have enough information to determine whether any four of these vectors form a basis for R4. Additionally, the statement in option d) does not mention any specific four vectors, so we cannot determine its validity based on the information provided.
Conclusion:
Based on the analysis of each statement, the false statement is option d) "Any four of these vectors form a basis for R4."
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If v1,v2, v6 are six vectors in R4, which one of the following statements is False?a)If {v1 v3, v5, v6} spans R4, then it forms a basis for R4.b)These vectors are not linearly independent.c)It is not necessary that these vectors span R4.d)Any four of these vectors form a basis for R4.Correct answer is option 'D'. Can you explain this answer?
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