Question

Consider the homogeneous system of linear equations: \( \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \). What does the solution set of this system represent geometrically?

• A. A point
• B. A line
• C. A plane
• D. A volume

Hint:

For a system \(Ax=0\) with the matrix \(A\) having \(m\) rows and \(n\) columns, the dimension of the solution space is \(n - \text{rank}(A)\). For \(n=3\), a dimension of 0 is a point (the origin), a dimension of 1 is a line, and a dimension of 2 is a plane.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the geometric interpretation of the solution set of a homogeneous system of linear equations \(Ax = 0\). The solution set forms a vector space called the null space of the matrix A.
Step 2: Geometric Interpretation:
The system of equations can be written as: \[ x_1 + x_2 + x_3 = 0 \] \[ x_1 + 2x_3 = 0 \] Each of these equations represents a plane in three-dimensional space (\(\mathbb{R}^3\)). The solution set of the system is the set of all points \((x_1, x_2, x_3)\) that lie on both planes simultaneously. In other words, the solution is the intersection of the two planes.
The normal vector of the first plane is \(\vec{n_1} = (1, 1, 1)\).
The normal vector of the second plane is \(\vec{n_2} = (1, 0, 2)\).
Since the normal vectors are not scalar multiples of each other, the planes are not parallel. The intersection of two non-parallel planes in 3D space is a line.
Step 3: Analysis using Rank-Nullity Theorem:
The coefficient matrix is \( A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{pmatrix} \). The number of variables is \(n = 3\). The rank of the matrix A is the number of linearly independent rows (or columns). The two rows are not multiples of each other, so the rank is 2. The Rank-Nullity Theorem states that \(\text{rank}(A) + \text{nullity}(A) = n\), where nullity is the dimension of the null space (the solution set). \[ 2 + \text{nullity}(A) = 3 \] \[ \text{nullity}(A) = 1 \] A one-dimensional subspace of \(\mathbb{R}^3\) is a line passing through the origin.
Step 4: Final Answer:
Both the geometric and algebraic analyses show that the solution set is a line.