Question

(1 point) Which of the following statements are true?A.The equation Ax=b is referred to as a vector equation.B.If the augmented matrix [ A b ] has a pivot position in every row, then the equation Ax=b is inconsistent.C.The first entry in the product Ax is a sum of products.D.If the columns of an m×n matrixA span Rm, then the equationAx=b is consistent for each b in Rm.E.The solution set of a linear system whose augmented matrix is [ a1 a2 a3 b ] is the same as the solution set of Ax=b, if A=[ a1 a2 a3 ].F.If A is an m×n matrix whose columns do not span Rm, then the equation Ax=b is inconsistent for some b in Rm.

Answer 1

Answer:

A. False

B. False

C. True

D. True

E. True

F. True

Explanation:

A. The equation Ax=b is referred to as a matrix equation and not vector equation.

B. If the augmented matrix [ A b ] has a pivot position in every​ row then equation Ax=b may or may not be consistent. It is inconsistent if [A b] has a pivot in the last column b and it is consistent if the matrix A has a pivot in every row.

C. In the product of Ax also called the dot product the first entry is a sum of products. For example the the product of Ax where A has [a11 a12 a13] in the first entry of each column and the corresponding entries in x are [x1 x2 x3] then the first entry in the product is the sum of products i.e.  a11x1 + a12x2 +a13x3

D. If the columns of mxn matrix A span R^m, this states that every possible vector b in R^m is a linear combination of the columns which makes the equation consistent. So the equation Ax=b has at least one solution for each b in R^m.

E. It is stated that a vector equation x1a1 + x2a2 + x3a3 + ... + xnan = b has the same solution set as that of the linear system with augmented matrix [a1 a2 ... an b]. So the solution set of linear system whose augmented matrix is [a1 a2 a3 b] is the same as solution set of Ax=b if A=[a1 a2 a3]  and b can be produced by linear combination of a1 a2 a3 iff the solution of linear system corresponding to [a1 a2 a3 b] takes place.

F.  It is true because lets say b is a vector in R^m which is not in the span of  the columns. b cannot be obtained for some x which belongs to R^m as b = Ax. So Ax=b is inconsistent for some b in R^m and has no solution.