Answer
a. False
b. False
c. True
d. False
e. False
Work Step by Step
a. For a single vector to be linearly dependent by itself, it would need to be the 0 vector. All other vectors are linearly independent.
b. The vectors in the set need to be linearly independent for them to form a basis for H.
c. An invertible matrix has a pivot in each row and column. This means the columns are linearly independent and do indeed form a basis for $R^n$.
d. A basis is a spanning set that is as small as possible.
e. Row operations do not change the linear dependence relation among columns. This is why we can use the reduced echelon form of a matrix to determine pivot columns.
