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

In Exercises 21 and 22, mark each statement True or False. Justify each answer. 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 $\operatorname{Col} A .$

In Exercises 21 and 22, mark each statement True or False. Justify each answer.
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 $\operatorname{Col} A .$

Key Concepts

Echelon Form and Pivot Columns

The echelon form of a matrix is obtained through row reduction and highlights the leading entries in each row, known as pivot positions. The pivot columns in the echelon form correspond to a basis for the column space of the original matrix, illustrating the relationship between row operations and the structure of a vector space. This technique is fundamental in linear algebra for determining linear independence and the span of the columns.

Maximal Linearly Independent Set

A maximal linearly independent set is one that is as large as possible while still maintaining linear independence. In finite-dimensional spaces, any maximal linearly independent set automatically spans the space, and therefore, it is a basis. This concept is essential for understanding how expansion by additional vectors will introduce dependencies.

Spanning Set

A spanning set of a vector space is a collection of vectors such that any vector in the space can be expressed as a linear combination of those vectors. It is crucial to understanding how vectors can be combined to cover an entire vector space or subspace.

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be written as a linear combination of the others. This concept is fundamental in determining whether a set of vectors can effectively represent unique directions in a vector space or subspace.

Basis

A basis is a set of vectors in a vector space that is both linearly independent and spanning. This means every vector in the space can be uniquely represented as a linear combination of the basis vectors. The concept of a basis emphasizes the minimal and complete nature required for uniquely synthesizing the vector space.

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