Problem 2. (4 points) Given that A is a matrix of size 6 by 5. Which of the following statements must be TRUE? (I) The linear system Ax = b is consistent for all b in R 6 . (II) The linear system Ax = b is consistent for some b in R 6 . (III) If the linear system Ax = b is consistent for a fixed b in R 6 , then its solution must not be unique. (A) I only (B) II only (C) III only (D) II and III only (E) I and II

Step-by-step explanation:

If the columns span R^6, this says that every b in R^6 is in the span of the columns, which is another way of saying that any b is a linear combination of the columns. Then the equation is consistent for some b or all b in R^6.

If an augmented matrix can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.

Any matrix can be transformed by elementary row operations into reduced echelon form, but not every matrix equation Ax = b is consistent

For example, after echelon reduction we have

1 6. 7. 8. 8.

0 1. 5. 4. 9

0. 0. 1. 3. 6

0. 0. 0. 1. 6.

0. 0. 0. 0. 1

0. 0. 0. 0. 0

In this case, the matrix doesn't have a solution, then the matrix is inconsistent

But if the matrix echelon reduction gives

1 6. 7. 8. 8.

0 1. 5. 4. 9

0. 0. 1. 3. 6

0. 0. 0. 1. 6.

0. 0. 0. 0. 1

0. 0. 0. 0. 1

Then, the system have a solution but not unique, the system have a solution for some b in R^6.

Then, the solution is II and III

So the correct option is D.