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: Determine Rank.
The matrix $A$ has dimensions $2 \times 3$. The two row vectors $[1, 1, 1]$ and $[1, 0, 2]$ are linearly independent because one is not a scalar multiple of the other. Thus, $\text{Rank}(A) = 2$.
Step 2: Apply Rank-Nullity Theorem.
$\text{Dimension of Null Space} = n - \text{Rank}(A) = 3 - 2 = 1$. The solution to a homogeneous system $Ax = 0$ is the Null Space.
Step 3: Geometric Interpretation.
A 1-dimensional subspace in $\mathbb{R}^3$ corresponds to a straight line passing through the origin.