Question

Let \( v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) and \( v_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \). Let \(M\) be the matrix whose columns are \(v_1,\; v_2,\; 2v_1 - v_2,\; v_1 + 2v_2\) in that order. Then the number of linearly independent solutions of the homogeneous system \(Mx = 0\) is ...........

Hint:

To find the number of linearly independent solutions to a homogeneous system, use the rank-nullity theorem: \( \text{nullity}(M) = n - \text{rank}(M) \).

Answer 1

Correct Answer: 1.9 - 2.1

Step 1: Construct the matrix M.
The matrix M has columns:

\[ M = \begin{bmatrix} 1 & 0 & 2 & 1 \\ 1 & 1 & -1 & 2 \end{bmatrix} \]
Step 2: Analyzing the rank of M.
We perform Gaussian elimination to find the rank of the matrix. After reducing the matrix, we find that the rank of M is 2.

Step 3: Determining the number of linearly independent solutions.
By the rank–nullity theorem:

\[ \text{nullity}(M) = 4 - \text{rank}(M) = 4 - 2 = 2 \]
Step 4: Conclusion.
Thus, the number of linearly independent solutions is \[ \boxed{2} \]