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?
Correct Answer: -1 - 1
Solution and Explanation
We are given two vectors that belong to the null space of a matrix \(A\) of rank 2. By the rank–nullity theorem, the dimension of the null space is:
\[ \text{nullity}(A) = 4 - 2 = 2 \]
Hence, the null space consists of exactly two linearly independent vectors. Any vector in the null space must be a linear combination of the given two vectors:
\[ \begin{bmatrix}2 \\ 1 \\ 0 \\ 3\end{bmatrix}, \quad \begin{bmatrix}1 \\ 0 \\ 1 \\ 2\end{bmatrix} \]
Check Option A:
Test whether the vector
\[ \begin{bmatrix}1 \\ 1 \\ -1 \\ 1\end{bmatrix} \]
can be expressed as a linear combination of the given vectors. Assume:
\[ c_1 \begin{bmatrix}2 \\ 1 \\ 0 \\ 3\end{bmatrix} + c_2 \begin{bmatrix}1 \\ 0 \\ 1 \\ 2\end{bmatrix} = \begin{bmatrix}1 \\ 1 \\ -1 \\ 1\end{bmatrix} \]
This leads to the system of equations:
\[ 2c_1 + c_2 = 1 \] \[ c_1 = 1 \] \[ c_2 = -1 \] \[ 3c_1 + 2c_2 = 1 \]
The values \(c_1 = 1\) and \(c_2 = -1\) satisfy all equations.
Therefore, the vector \( \begin{bmatrix}1 \\ 1 \\ -1 \\ 1\end{bmatrix} \) is a linear combination of the given null-space vectors and hence belongs to the null space.
Final Answer:
\(\boxed{A}\)
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