Question

If \({\mathop{\rm b}\nolimits} \ne 0\), can the solution set of \(Ax = b\) be a plane through the origin? Explain.

Short answer

When \(b \ne 0\), the solution set of \(Ax = b\) cannot form a plane through the origin

Step-by-step solution

Determine whether the solution set of \(Ax = b\) is a plane through the origin

When \(b \ne 0\), the solution set of \(Ax = b\) cannot form a plane through the origin.

 Step 2: Explain that the solution set of \(Ax = b\) cannot be a plane through the origin

When \(b \ne 0\), the solution set of \(Ax = b\)cannot form a plane through the origin.

If the solution set of \(Ax = b\) contained the origin, then \(0\) would satisfy \(A0 = b\). This is not true because \(b\) is not the zero vector.