Final answer:
The columns of the matrix [-4, -3, 0; 0, -1, 4; 1, 0, 3; 5, 4, 6] form a linearly independent set.
Step-by-step explanation:
To determine if the columns of a matrix form a linearly independent set, we need to check if the only solution to the equation Ax = 0, where A is the given matrix and x is a vector, is the trivial solution x = 0.
In this case, the given matrix is [-4, -3, 0; 0, -1, 4; 1, 0, 3; 5, 4, 6]. To check if the columns are linearly independent, we can set up the equation Ax = 0:
-4x1 + 0x2 + 1x3 + 5x4 = 0
-3x1 - 1x2 + 0x3 + 4x4 = 0
0x1 + 4x2 + 3x3 + 6x4 = 0
We can solve this system of equations using Gaussian elimination or by using the matrix inverse. If the only solution is x = 0, then the columns are linearly independent. If there are other non-zero solutions, then the columns are linearly dependent.
Performing Gaussian elimination on the given matrix, we obtain:
[1, 0, 0; 0, 1, 0; 0, 0, 1; 0, 0, 0]
Since there are no free variables and the only solution is x = 0, we can conclude that the columns of the matrix[-4, -3, 0; 0, -1, 4; 1, 0, 3; 5, 4, 6] form a linearly independent set.