Answer
a. True
b. False
c. True
d. True
e. True
Work Step by Step
a. True
b. Because a line can be represented by 1 vector, it is a one dimensional subspace of $R^n$ if it passes through the origin.
c. Col A is the set of all linear combinations of A. Its basis, however, consists of the linearly independent vectors in A. Because the number of linearly independent vectors is equal to the number of pivot columns, this is true.
d. The dimensions of Nul A is equal to the number of free variables in the equation $Ax=0$. The sum of linearly independent columns and free variables is the number of columns in A
e. If a set of p vectors span a p dimensional subspace, the matrix whose columns are each vector in p has p rows that contain a pivot. Therefore, these are linearly independent and form a basis for H.
