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Werg22

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- Thread starter Werg22
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In summary, if a finite set S is linearly independent, this implies the existence of a set T of the same size where the elements of T are pairwise orthogonal. This can be found using methods such as Gram-Schmidt and it is likely that the basis spanning T will also be orthogonal to the basis spanning S.

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Werg22

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A KillEase

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If it's in an inner product space, you can use Gram-Schmidt to find T.

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intrepid_nerd

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Werg22 said:

yes. i believe any method you use (eigenspace, nullspace, etc) to find a set that spans T will be mutually orthogonal. i'd probably guess that the basis spanning T would be orthogonal to the basis spanning A as well.

Linear independence is a mathematical concept that describes the relationship between vectors. It means that a set of vectors is considered linearly independent if none of the vectors can be written as a linear combination of the others.

To determine if a set of vectors is linearly independent, you can use the following criteria: If the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0, then the vectors are linearly independent.

Orthogonality is a mathematical concept that describes the relationship between vectors. It means that two vectors are orthogonal if their dot product is equal to 0. In other words, the angle between the two vectors is 90 degrees.

To determine if two vectors are orthogonal, you can use the dot product formula: v1 · v2 = |v1| * |v2| * cosθ, where θ is the angle between the two vectors. If the dot product is equal to 0, then the vectors are orthogonal.

Linear independence and orthogonality are closely related concepts. In fact, if a set of vectors is both linearly independent and orthogonal, then it is also considered to be a basis for the vector space. This means that the vectors can be used to uniquely represent any vector in the space. Additionally, if a set of vectors is orthogonal but not linearly independent, it can be transformed into a linearly independent set by normalizing the vectors (making them unit vectors).

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