If the linear system is inconsistent, then b is a linear combination of v1, v2, ..., vp.*TrueFalse

The statement is False.

Explanation:

1. Definition of Inconsistency: A linear system is considered inconsistent if there is no solution, meaning that the equations represented by the system do not intersect at any point in the vector space.

2. Linear Combination Concept: A vector $b$ can be expressed as a linear combination of $v_1, v_2, \ldots, v_p$ if there exist scalars $c_1, c_2, \ldots, c_p$ such that: $b = c_1 v_1 + c_2 v_2 + \ldots + c_p v_p$

3. Relation: If the linear system is inconsistent, it means that $b$ cannot be obtained from the linear combination of the vectors $v_1, v_2, \ldots, v_p$ spanning the space defined by the equations. Hence, $b$ is not a linear combination of $v_1, v_2, \ldots, v_p$.

Conclusion

Since an inconsistent system indicates that the vector $b$ lies outside the span of $v_1, v_2, \ldots, v_p$, the statement that $b$ is a linear combination of these vectors is indeed false.