Question

Let 

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 of linear equations \( Mx = 0 \) 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

Step 1: Construct the matrix \( M \). 
The matrix \( M \) has columns: 

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, the number of linearly independent solutions of the homogeneous system \( Mx = 0 \) is the nullity of \( M \), which is: \[ \text{nullity}(M) = 4 - \text{rank}(M) = 4 - 2 = 2. \] 
Step 4: Conclusion. 
Thus, the number of linearly independent solutions is \( \boxed{2} \).