Question 1. A vector b is a linear combination of the columns of a matrix A is and only if the equation Ax = b has at least one solution.
Answer: True. These are two ways of saying the same thing.
Question 2. The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row.
Answer: False. The system is inconsistent if [A b] has a pivot in the last ("b") column. The system is consistent if the matrix A has a pivot in every row.
Question 3. If the columns of an mxn matrix A span R^m, then the equation
Ax = b is consistent for each b in R^m.
Answer: True. If the columns span R^m, this says that every b in R^m 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 (see Question 1).
Question 4. If the equation Ax = b is inconsistent, then b is not in the set spanned by the columns of A.
Answer: True.
Question 5. The solution set of a linear system whose augmented matrix is [a b c d] is the same as the solution set of Ax = d, where A = [a b c]. Note: a, b, c, d are all column vectors.
Answer: True.
Question 6. Let A be a 3x4 matrix and let y1 and y2 be vectors in R^3, with w = y1 + y2. Suppose that y1 = Ax1 and y2 = Ax2 for some vectors x1 and x2 in R^4. Which of the following is true?
Answer: Let x = x1+x2. Then Ax = A(x1+x2) = Ax1 + Ax2 = y1 +y2 by the distributive property of matrix multiplication (see problem 35). Thus x is a solution of Ax = y, with no further information needed. However x is not necessarily the general solution, since Ax = y may have an infinite number of solutions.