2 Suppose that A is an m n matrix a How do we determine the

2. Suppose that A is an m × n matrix.

(a) How do we determine the number of pivotal columns?

(b) What do the pivotal columns tell us about the solution to the equation A!x =!b?

(c) What space is equal to the span of the pivotal columns?

(d) Is it possible to write A = LU, where L is a lower triangular matrix and U is an upper triangular matrix? If not, what additional information would you need?

(e) Is it possible to write A = QR, where Q has orthonormal columns and R is an upper triangular matrix? If not, what additional information would you need?

(f) What is the difference between solving A!x = !b and A!x = !0? How are these two solutions related geometrically?

(g) If rank(A) = r, where 0 < r ? n, how many columns are pivotal? What is the dimension of the solution space to A!x = !0?

2. Suppose that A is an m x n matrix (a) How do we determine the number of pivotal columns? (b) What do the pivotal columns tell us about the solution to the equation A (c) What space is equal to the span of the pivotal columns? (d) Is it possible to write A = LU, where L is a lower triangular matrix and U is an upper triangular m at (e) Is it possible to write A = QR, where Q has orthonormal columns and R is an upper triangular m at (f) What is the difference between solving Ax = b and Ax-0? How are these two solutions related geometrically? (g) If rank(A) r, where 0 r n, how many columns are pivotal? What is the dimension of the solution space to Ai - 0? atrix? If not, what additional inform ion would you need? atrix? If not, what additional inform ion would you need?

Solution

The pivotal columns are the columns that contain the leading 1’s of the rows. The pivotal columns form the basis of the column space of the matrix. The number of pivotal columns is always equal to the number of non-zero rows in the RREF of the matrix.