Efficient Method for Finding Basis and Determinant of 4 Vectors in Matrix Form

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In summary, the purpose of finding the basis and determinant of 4 vectors in matrix form is to determine the linear independence of the vectors and the volume of the parallelepiped formed by them. The basis is found by identifying a set of linearly independent vectors that span the same space as the original 4 vectors. The determinant is a scalar value that represents the volume of the parallelepiped and is calculated using the cross and dot products of the vectors. The significance of the determinant is in solving linear systems of equations and in calculating volumes and areas in physics and engineering. There are efficient methods, such as row reduction and Gaussian elimination, for finding the basis and determinant of 4 vectors in matrix form.
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pyroknife
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I know how to do the problem, just put the 4 vectors in matrix form and find for what values of k is the detminant =0. the answer is then that k can't equal the value that was found.

Is there a easier way to do this?

My method involves finding the determinant using the expansion method, which seems like a long way. Is there a faster way?
 

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In this example, it's quicker to spot simplifications first.
Add 3rd vector to 2nd to produce (0, 2, 0, 7).
Multiply 1st by 7 and new 2nd by 2 then subtract 2nd from 1st.
etc.
 

FAQ: Efficient Method for Finding Basis and Determinant of 4 Vectors in Matrix Form

What is the purpose of finding the basis and determinant of 4 vectors in matrix form?

The basis and determinant of 4 vectors in matrix form help to determine the linear independence of the vectors and the volume of the parallelepiped formed by the vectors. This information is crucial in solving linear systems of equations and in applications such as physics and engineering.

How is the basis of 4 vectors in matrix form determined?

The basis of 4 vectors in matrix form is determined by finding a set of linearly independent vectors that span the same space as the original 4 vectors. This can be done by row reducing the matrix and identifying the pivot columns.

What is the determinant of 4 vectors in matrix form?

The determinant of 4 vectors in matrix form is a scalar value that represents the volume of the parallelepiped formed by the 4 vectors. It is calculated by taking the cross product of any two vectors and finding the dot product with the remaining two vectors.

What is the significance of the determinant of 4 vectors in matrix form?

The determinant of 4 vectors in matrix form is important in determining if the vectors are linearly independent, which is necessary for solving linear systems of equations. It is also used in calculating volumes and areas in physics and engineering applications.

Is there an efficient method for finding the basis and determinant of 4 vectors in matrix form?

Yes, there are various efficient methods for finding the basis and determinant of 4 vectors in matrix form, such as row reduction, Gaussian elimination, and using the properties of determinants. These methods can be easily implemented using computer software or calculators.

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