Answer:
A. False
B. False
C. True
D. True
E. True
F. True
Explanation:
A. The term Ax=b describes a matrix equation rather than a vector equation.
B. An augmented matrix [ A b ] having a pivot in every row does not guarantee that the equation Ax=b is consistent; it may be inconsistent if there is a pivot in the last column b, but is consistent if matrix A has a pivot in every row.
C. When considering the product Ax, also known as the dot product, the first element is derived from the sum of products. For instance, in the case where A has [a11 a12 a13] as the first elements in its columns, with corresponding entries in x being [x1 x2 x3], the first element of the product becomes a11x1 + a12x2 + a13x3.
D. If an m×n matrix A spans R^m, it indicates that every possible vector b in R^m can be expressed as a linear combination of the columns, making the equation consistent. This means Ax=b has at least one solution for every b in R^m.
E. A vector equation expressed as x1a1 + x2a2 + x3a3 + ... + xnan = b shares the same solution set as the linear system represented by the augmented matrix [a1 a2 ... an b]. Thus, the solution set of the linear system indicated by [a1 a2 a3 b] coincides with the solution set of Ax=b, provided A=[a1 a2 a3] and b can result from a linear combination of a1, a2, a3, if the linear system corresponding to [a1 a2 a3 b] is satisfied.
F. It is accurate to assert that if b is a vector in R^m that does not lie within the span of the columns, then there exists no value of x within R^m such that b = Ax. Consequently, Ax=b becomes inconsistent for particular b in R^m, indicating it lacks a solution.
