Question

Every finite dimensional inner product space has an orthogonal basis

Answer 1

Every finite-dimensional inner product space has an orthogonal basis - True or false

Answer:

The statement that every finite-dimensional inner product space has an orthogonal basis is true and can be proven using the Gram-Schmidt process.

Explanation:

The Gram-Schmidt process is a method for constructing an orthogonal basis from a given set of linearly independent vectors. Starting with a basis for the inner product space, the process generates an orthonormal basis by successively subtracting the projections of each vector from the previous ones and normalizing the result.

By repeating this process until all vectors in the original basis have been exhausted, we obtain an orthonormal basis that spans the entire space. This is possible because the Gram-Schmidt process guarantees that each new vector generated is orthogonal to all previous vectors, and each normalized vector has a unit length.

Since every finite-dimensional inner product space has a finite basis, it follows that such a space can always be equipped with an orthogonal basis.

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