Gram-Schmidt

Given any finite dimensional inner product space, it is possible to find an orthonormal basis.

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Any n dimensional real inner product space is isomorphic (there exists a linear transformation that preserves inner products) to  $\mathbb{R}^n$  under the dot product.

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Every inner product space has an orthonormal basis via the Gram Schmidt process.

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If we start with a set of linearly dependent vectors, the Gram Schmidt process will eventually produce the zero vector.

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The set $1,\sqrt{3}\left(2x-1\right),\sqrt{5}\left(6x^2+6x+1\right)$  is an orthonormal basis for the set of continuous functions on [0,1].

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The set $1,\ \sqrt{3}\left(2x-1\right),\ \sqrt{5}\left(6x^2-6x+1\right)$  is the unique orthonormal basis starting from the standard basis on the set of polynomials of degree 2.

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The same linearly independent set written in a different order will in general produce a different orthonormal basis.

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Given any basis for a finite dimensional vector space, it is possible to define an inner product for which it is an orthonormal basis.

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The big consequence of the Gram Schmidt Theorem is the fact that all finite dimensional inner product spaces are isomorphic to either  $\mathbb{R}^n$  under the dot product or  $\mathbb{C}^n$  under the standard Hermitian inner product.

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There exists an inner product that is always zero.

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• False