Question-3

Select all correct options concerning a system of linear equations given by \(Ax = b\), where \(A\) is a \(m \times n\) matrix, \(x \in \mathbb{R}^{n}\) and \(b \in \mathbb{R}^{m}\).

• The solution to this system is given by \(A^{-1} b\).

• The system has a solution if and only if \(b\) is one of the columns of \(A\).

• The system has a solution if and only if \(b\) is in the column-space of \(A\).

• If this system is consistent, it can have either a unique solution or infinitely many solutions.

Answer

• The solution to this system is given by \(A^{-1} b\).

• The system has a solution if and only if \(b\) is one of the columns of \(A\).

• The system has a solution if and only if \(b\) is in the column-space of \(A\).

• If this system is consistent, it can have either a unique solution or infinitely many solutions.

Solution

• The system is consistent if \(b\) can be expressed as a linear combination of the columns of \(A\).
• A consistent system that has at least two solutions cannot have finitely many solutions. To see why this is true, let \(x_1\) and \(x_2\) be two solutions to \(Ax = b\) with \(x_1 \neq x_2\). Then \((a + 1)x_1 - ax_2\) is also a solution for \(a \in \mathbb{N}\). Using this, one can generate infinitely many solutions.