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 / times n matrix form a basis for R^n.

d. In some cases, the linear dependence relations amoung 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 large as possible.

a) False

b) False

c) True

d) False

e) False

Step-by-step explanation:

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

If v = 0 then the only scalar c such that cv = 0 is c = 0. Hence, 1vl is linearly independent. A set consisting of a single vector v is linearly dependent if and only if v = 0. Therefore, only a single zero vector is linearly dependent, while any set consisting of a single nonzero vector is linearly independent.

b. If H = Span{b1, ... bp}, then {b1, ... bp} is a basis for H. False

A sets forms a basis for vector space, only if it is linearly independent and spans the space. The fact that it is a spanning set alone is not sufficient enough to form a basis.

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

If a matrix is invertible, then its columns are linearly independent and every row has a pivot element. The columns, can therefore, form a basis for Rⁿ.

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. False

Row operations can not affect linear dependence among the columns of a matrix.

e. A basis is a spanning set that is as large as possible. False

A basis is not a large spanning set. A basis is the smallest spanning set.