Determine if each statement a. through e. below is true or false. Justify each answer. a. A linearly independent set in a subspace H is a basis for H.

1. The statement is true by the Spanning Set Theorem
2. The statement is false because the set must be linearly dependent.
3. The statement is false because the subspace spanned by the set must also coincide with H
4. The statement is true by the definition of a basis

b. If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V

1. The statement is false because the subspace spanned by the set must also coincide with V
2. The statement is false because the subset must be independent.
3. The statement is true by the Spanning Set Theorem.
4. The statement is true by the definition of a basis

c. A basis is a linearly independent set that is as large as possible

1. The statement is false because a basis is a linearly dependent set.
2. The statement is true by the Spanning Set Theorem
3. The statement is false because a basis is the smallest independent set that spans the subspace.
4. The statement is true by the definition of a basis

d. The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.

1. The statement is true because the set produced may not be independent.
2. The statement is false because the method always produces an independent set.
3. The statement is true because the only set produced may be the trivial solution.
4. The statement is false because a spanning set for Nul A also spans A.

e. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.

1. The statement is true by the Invertible Matrix Theorem
2. The statement is true by the definition of a basis ° C.
3. The statement is false because the columns of an echelon form B of A are not necessarily in the column space of A.
4. The statement is false because the pivot columns of A form a basis for Col B