Question

In Exercises 21 and 22, mark each statement True or False. Justify

each answer.

22. a.A linearly independent set in a subspace H is a basis for H.

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

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

d. The standard method for producing a spanning set for Nul A, described in Section 4.2, sometimes fails to produce a basis for Nul A.

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

Short answer

1. The statement is false.
2. The statement is true.
3. The statement is true.
4. The statement is false.
5. The statement is false.

Step-by-step solution

(a) Step 1: Mark the first statement true or false

The statement “A linearly independent set in a subspace H is a basis for H.” cannot be true. The reason is mentioned below:

The set is a basis for H if the set of vectors is linearly independent and also spans the space.

Thus, statement (a) is false.

Mark the second statement true or false

The given statement “If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.” is true.

The reason is mentioned below:

Suppose the set of nonzero vectors\(S = \left\{ {{{\bf{v}}_1},{{\bf{v}}_2},...,{{\bf{v}}_p}} \right\}\)is in vector space V. From set S, any one of the vectors is a linear combination of other vectors.

Since set S uses nonzero vectors, space V is nonzero. This satisfies the spanning set theorem.

Thus, statement (b) is true.

(c) Step 3: Mark the third statement true or false

Suppose the set of nonzero vectors\(S = \left\{ {{{\bf{v}}_1},{{\bf{v}}_2},...,{{\bf{v}}_p}} \right\}\)is in vector space V. From set S, any one of the vectors is a linear combination of other vectors.

Since set S uses nonzero vectors, space V is nonzero, which satisfies the spanning set theorem.

The number of spanning sets can be more than the number of basis sets in\({\mathbb{R}^n}\). Not every spanning set can be used as a basis. So, the set is as large as possible.

Thus, statement (c) is true.

(d) Step 4: Mark the fourth statement true or false

The conventional method for determining the basis of Nul A gives linearly independent results if the null space of matrix A contains nonzero vectors. So, it always produces a basis for Nul A.

Thus, statement (d) is false.

(e) Step 5: Mark the fifth statement true or false

The statement “If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.” cannot be true.

The pivot columns of B show which columns of A are used as a basis for the column space of matrix A. So, the column space of matrix A does not use the columns of B as a basis.

Thus, statement (e) is false.