Check the true statements below: a. A single vector by itself is linearly dependent. b. If H = Span{b1,...,bp}, then {b1,...,bp} is a basis for H. c. The columns of an invertible n x n matrix form a basis for R^n. d. In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix. e. A basis is a spanning set that is as small as possible.

a. True. A single vector is linearly dependent if it is the zero vector. If it is a non-zero vector, it is linearly independent, as there is no nontrivial linear combination (combination where not all the coefficients are zero) that will result in the zero vector; but since the decision of linear dependence does not require a nontrivial combination, even a single vector \( \vec{v} \) is considered linearly dependent as \( 0\vec{v} = \vec{0} \).

b. False. If H = Span{b1,...,bp}, then {b1,...,bp} could be a basis for H if the vectors are linearly independent. If the vectors are not linearly independent, then they do not form a basis for H.

c. True. The columns of an invertible n x n matrix are linearly independent, and they span R^n (since the matrix is invertible, it must transform R^n onto itself). Therefore, the columns form a basis for R^n.

d. False. Elementary row operations do not affect the linear dependence relations among the columns of a matrix. They may change the actual values within the matrix, but not the fundamental linear relationships among columns.

e. True. A basis is indeed a spanning set that is as small as possible. This means that it is a set of vectors that span a space with no redundancies—it is linearly independent.