Chapter 4.3, Problem 21E

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

21. a. A single vector by itself is linearly dependent.

b. If H = Span {b₁,…,b_p}, then {b₁,…,b_p} is a basis for H.

c. The columns of an invertible n × n matrix form a basis for ℝⁿ.

d. A basis is a spanning set that is as large as possible.

e. In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.