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Determine if the columns of the matrix form a linearly independent set. Justify your answer. [-4, -3, 0; 0, -1, 4; 1, 0, 3; 5, 4, 6]

User Vao Tsun
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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.

User Deependra Singh
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