Question

Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. Choose the correct answer. 0

A. Since Ax = b is inconsistent, its solution set is obtained by translating the solution set of Ax-0. For Ax-b to be inconsistent, Ax = 0 has only the trivial solution.
B. Since Ax-b is consistent, its solution set is obtained by translating the solution set of Ax-0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax 0 is a single vector, and that happens if and only if Ax 0 has only the trivial solution.
C. Since Ax = b is inconsistent, then the solution set of Ax = 0 is also inconsistent. The solution set of Ax = 0 is inconsistent if and only if Ax = 0 has only the trivial solution.
D. Since Ax = b is consistent, then the solution is unique if and only if there is at least one free variable in the corresponding system of equations. This happens if and only if the equation Ax 0 has only the trivial solution.

Answer 1

B. Since Ax-b is consistent, its solution set is obtained by translating the solution set of Ax=0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax= 0 is a single vector, and that happens if and only if Ax 0 has only the trivial solution.

Explanation:

the answer to the question is answer B. and here is the explanation below

let us imagine that the equation ax = b has a solution

now our goal will be to show that the solution of ax =b when ax = 0 has only trivial solution.

ax = 0 is homogenous

if this equation was consistent for b, we define

ax = b to be a set of vector that has the form

w = m + gh(h is a subscript)

gh is a solution of ax = 0

from what we have above, ax=b is in the form ofw= m+gh

with

m = solution of ax=b

gh = soulution of ax=0

ax = 0 has only trivial solution

gh = 0

with gh = 0

ax=b is w=m

so ax = b is unique.