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# Orthogonality.

Definition 12.3.1   Let be an vector space with an inner product. Two vectors and are orthogonal if their inner product is equal to 0.

Example 12.3.2   Let with the regular'' inner product defined in Example  1.3.
• Take and . Then we have: , i.e. the vectors and are orthogonal.
• Take and . Then we have: , i.e. the vectors and are not orthogonal.
• The vector of the standard basis of are orthogonal.

Example 12.3.3   Let be the space of all the polynomials in one variable x with coefficients in , with inner product:

If f1(x)=x and f2(x)=x2, then . Thus the functions f1 and f2 are not orthogonal vectors in .

Example 12.3.4   Let , with the inner product

Then the sine and the cosine function are orthogonal.

Noah Dana-Picard
2001-02-26