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cscott
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Will a set of vectors stay linearly independent after a change of basis? If it's not always true then is it likely or would you need a really contrived situation?
Linear independence after change of basis refers to the property of a set of vectors to remain linearly independent even after a change of basis. This means that the vectors are still unique and cannot be expressed as a linear combination of each other, even when the coordinate system is changed.
Understanding linear independence after change of basis is important because it allows us to manipulate and transform vectors and matrices without losing their independence. This is especially useful in applications such as linear transformations and change of coordinate systems.
The criteria for linear independence after change of basis are the same as for standard linear independence. The vectors must be unique and cannot be expressed as a linear combination of each other. In other words, the determinant of the matrix formed by the vectors must be non-zero.
To determine if a set of vectors is linearly independent after a change of basis, we can perform a change of basis using the matrix formed by the original and new basis vectors. If the determinant of this matrix is non-zero, then the vectors are still linearly independent. If the determinant is zero, then the vectors are linearly dependent after the change of basis.
Some practical applications of understanding linear independence after change of basis include image processing, signal processing, and data compression. These fields often involve transforming and manipulating data in different coordinate systems, and understanding linear independence allows for accurate and efficient calculations.