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In the statements below, V is a vector space. Mark each statement true or false. Justify each answer. C. A vector space infinite-dimensional if it is spanned by an infinite set. Choose the correct answer below. O A. True, because the dimension of a vector space is equal to the number of elements in a set that spans the vector space. O B. False, because all vector spaces are finite-dimensional. OC. False, because a basis for the vector space may have only finitely many elements, which would make the vector space finite-dimensional. O D. True, because the dimension of a vector space is the number of vectors in a basis for that vector space, and a vector space spanned by an infinite set has a basis with an infinite number of vectors. d. If dim V=n and if S spans V, then S is a basis of V. Choose the correct answer below. O A. False, in order for S to be a basis, it must also have n elements. O B. True, because if a vector space is finite-dimensional, then a set that spans it is a basis of the vector space. OC. False, because the set S must have less than n elements. O D. True, because if a set spans a vector space, regardless of the dimension of the vector space, then that set is a basis of the vector space. e. The only three-dimensional subspace of R3 is R3 itself. Choose the correct answer below. O A. False, because any subspaces of R3 which contain three-element vectors are three-dimensional, but most of these subspaces do not contain all of R3. O B. True, because any three linearly independent vectors in R3 span all of R3, so there is no three-dimensional subspace of R3 that is not R3 itself. OC. False, because most three-dimensional subspaces of R3 are spanned by a linearly dependent set of three vectors, but R3 can only be spanned by three linearly independent vectors. O D. True, because any three linearly dependent vectors in R3 span all of R?, so there is no three-dimensional subspace of R? that is not R3 itself.

In the statements below, V is a vector space. Mark each statement true or false. Justify each answer. C. A vector space infinite-dimensional if it is spanned by an infinite set. Choose the correct answer below. O A. True, because the dimension of a vector space is equal to the number of elements in a set that spans the vector space. O B. False, because all vector spaces are finite-dimensional. OC. False, because a basis for the vector space may have only finitely many elements, which would make the vector space finite-dimensional. O D. True, because the dimension of a vector space is the number of vectors in a basis for that vector space, and a vector space spanned by an infinite set has a basis with an infinite number of vectors. d. If dim V=n and if S spans V, then S is a basis of V. Choose the correct answer below. O A. False, in order for S to be a basis, it must also have n elements. O B. True, because if a vector space is finite-dimensional, then a set that spans it is a basis of the vector space. OC. False, because the set S must have less than n elements. O D. True, because if a set spans a vector space, regardless of the dimension of the vector space, then that set is a basis of the vector space. e. The only three-dimensional subspace of R3 is R3 itself. Choose the correct answer below. O A. False, because any subspaces of R3 which contain three-element vectors are three-dimensional, but most of these subspaces do not contain all of R3. O B. True, because any three linearly independent vectors in R3 span all of R3, so there is no three-dimensional subspace of R3 that is not R3 itself. OC. False, because most three-dimensional subspaces of R3 are spanned by a linearly dependent set of three vectors, but R3 can only be spanned by three linearly independent vectors. O D. True, because any three linearly dependent vectors in R3 span all of R?, so there is no three-dimensional subspace of R? that is not R3 itself.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In the statements below, V is a vector space. Mark each statement true or false. Justify each answer. C. A vector space is infinite-dimensional if it is spanned by an infinite set. Choose the correct answer below. O A. True, because the dimension of a vector space is equal to the number of elements in a set that spans the vector space. O B. False, because all vector spaces are finite-dimensional. O C. False, because a basis for the vector space may have only finitely many elements, which would make the vector space finite-dimensional. O D. True, because the dimension of a vector space is the number of vectors in a basis for that vector space, and a vector space spanned by an infinite set has a basis with an infinite number of vectors. d. If dim V=n and if S spans V, then S is a basis of V. Choose the correct answer below. O A. False, in order for S to be a basis, it must also have n elements. O B. True, because if a vector space is finite-dimensional, then a set that spans it is a basis of the vector space. OC. False, because the set S must have less than n elements. O D. True, because if a set spans a vector space, regardless of the dimension of the vector space, then that set is a basis of the vector space. e. The only three-dimensional subspace of R? is R3 itself. Choose the correct answer below. O A. False, because any subspaces of R3 which contain three-element vectors are three-dimensional, but most of these subspaces do not contain all of R3. O B. True, because any three linearly independent vectors in R3 span all of R3, so there is no three-dimensional subspace of R3 that is not R3 itself. O C. False, because most three-dimensional subspaces of R3 are spanned by a linearly dependent set of three vectors, but R3 can only be spanned by three linearly independent vectors. O D. True, because any three linearly dependent vectors in R3 span all of R3, so there is no three-dimensional subspace of R3 that is not R3 itself.
Transcribed Image Text:In the statements below, V is a vector space. Mark each statement true or false. Justify each answer. C. A vector space is infinite-dimensional if it is spanned by an infinite set. Choose the correct answer below. O A. True, because the dimension of a vector space is equal to the number of elements in a set that spans the vector space. O B. False, because all vector spaces are finite-dimensional. O C. False, because a basis for the vector space may have only finitely many elements, which would make the vector space finite-dimensional. O D. True, because the dimension of a vector space is the number of vectors in a basis for that vector space, and a vector space spanned by an infinite set has a basis with an infinite number of vectors. d. If dim V=n and if S spans V, then S is a basis of V. Choose the correct answer below. O A. False, in order for S to be a basis, it must also have n elements. O B. True, because if a vector space is finite-dimensional, then a set that spans it is a basis of the vector space. OC. False, because the set S must have less than n elements. O D. True, because if a set spans a vector space, regardless of the dimension of the vector space, then that set is a basis of the vector space. e. The only three-dimensional subspace of R? is R3 itself. Choose the correct answer below. O A. False, because any subspaces of R3 which contain three-element vectors are three-dimensional, but most of these subspaces do not contain all of R3. O B. True, because any three linearly independent vectors in R3 span all of R3, so there is no three-dimensional subspace of R3 that is not R3 itself. O C. False, because most three-dimensional subspaces of R3 are spanned by a linearly dependent set of three vectors, but R3 can only be spanned by three linearly independent vectors. O D. True, because any three linearly dependent vectors in R3 span all of R3, so there is no three-dimensional subspace of R3 that is not R3 itself.
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