Question

decide if each of the following statements is true or false. justify your conclusion with an explanation or counter-example, as appropriate. applying gram-schmidt to an orthogonal set of vectors returns the vectors unchanged.

Answer 1

The statement is true; applying Gram-Schmidt to an already orthogonal set of vectors will return the vectors unchanged because the process does not project out any components between orthogonal vectors.

Step-by-step explanation:

The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, which essentially makes a set of non-orthogonal vectors orthogonal.

However, if the set of vectors is already orthogonal, then each step of the Gram-Schmidt process will simply scale the existing vectors to turn them into unit vectors, if they are not already of unit length.

Since no components are projected out (as none are shared between the orthogonal vectors), the original set of vectors remains the same.

By definition, orthogonal vectors have a dot product of zero. Given two orthogonal unit vectors ei and ej, the scalar product is cos 90° = (1)(1)(0) = 0.

Thus, when applying the Gram-Schmidt process to these vectors, each step will not alter any vector, because each one is already orthogonal to the others in the set.