# Linear independence and orthogonaliy

• Werg22
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.
Werg22
Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?

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

Werg22 said:
Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?

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.

## 1. What is linear independence?

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.

## 2. How is linear independence determined?

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.

## 3. What is orthogonality?

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.

## 4. How is orthogonality determined?

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.

## 5. What is the relationship between linear independence and orthogonality?

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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