Which of the following is true?Question 5Select one:All are true.Two subsets of a vector space V that span the same subspace of V must be equal.The set of upper triangular n × n matrices is a subspace of the vector space of all n × n matrices.Every subset of a vector space V that contains the zero vector in V is a subspace of V.
Question
Which of the following is true?Question 5Select one:All are true.Two subsets of a vector space V that span the same subspace of V must be equal.The set of upper triangular n × n matrices is a subspace of the vector space of all n × n matrices.Every subset of a vector space V that contains the zero vector in V is a subspace of V.
Solution
The correct answer is: "The set of upper triangular n × n matrices is a subspace of the vector space of all n × n matrices."
Here's why:
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"Two subsets of a vector space V that span the same subspace of V must be equal." - This statement is false. Two subsets can span the same subspace without being equal. For example, the sets { (1,0), (0,1) } and { (1,0), (0,1), (1,1) } both span R^2, but they are not equal.
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"The set of upper triangular n × n matrices is a subspace of the vector space of all n × n matrices." - This statement is true. The set of upper triangular matrices is closed under addition and scalar multiplication, and contains the zero matrix, so it is a subspace.
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"Every subset of a vector space V that contains the zero vector in V is a subspace of V." - This statement is false. Just containing the zero vector does not make a set a subspace. The set also needs to be closed under addition and scalar multiplication. For example, the set { (0,0), (1,2) } in R^2 contains the zero vector, but is not a subspace because it is not closed under scalar multiplication.
Therefore, the statement "All are true" is also false.
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