Question

Two vectors \([2 \, 1 \, 0 \, 3]^T\) and \([1 \, 0 \, 1 \, 2]^T\) belong to the null space of a \(4 \times 4\) matrix of rank 2. Which one of the following vectors also belongs to the null space?

Hint:

To determine if a vector belongs to the null space of a matrix, check if the vector can be expressed as a linear combination of the known vectors that already belong to the null space. Solve the system of equations formed by the vectors to find the coefficients.

Answer 1

We are given two vectors that belong to the null space of a matrix \( A \) of rank 2. According to the rank-nullity theorem, the null space of a matrix with rank 2 will have a dimension of 2, meaning there are 2 linearly independent vectors in the null space. We are asked to identify which one of the given vectors also belongs to the null space. To do this, we need to check which vector is linearly dependent on the given two vectors. The vector will belong to the null space if it can be written as a linear combination of the given two vectors. Let's perform the necessary calculations for each option: Option A: \([1 \, 1 \, -1 \, 1]^T\) Check if this vector can be written as a linear combination of \([2 \, 1 \, 0 \, 3]^T\) and \([1 \, 0 \, 1 \, 2]^T\). We can solve this by checking the system of equations: \[ \text{Let} \, c_1 \cdot [2 \, 1 \, 0 \, 3]^T + c_2 \cdot [1 \, 0 \, 1 \, 2]^T = [1 \, 1 \, -1 \, 1]^T \] The system of equations becomes: \[ 2c_1 + c_2 = 1
c_1 = 1
c_2 = -1
3c_1 + 2c_2 = 1 \] Solving this system shows that \(c_1 = 1\) and \(c_2 = -1\), thus confirming that the vector \([1 \, 1 \, -1 \, 1]^T\) is a linear combination of the two given vectors. Thus, the correct answer is \(\boxed{A}\).