Question

Use the Gram-Schmidt process to transform the following sequence of vectors into an orthonormal sequence of vectors.

Answer 1

The Gram-Schmidt process is a method used to transform a sequence of vectors into an orthonormal sequence of vectors. It involves finding orthogonal vectors and normalizing them to obtain unit vectors.

Step-by-step explanation:

The Gram-Schmidt process is used to transform a sequence of vectors into an orthonormal sequence of vectors. The process involves finding a set of orthogonal vectors by subtracting the projection of one vector onto another from the original vector. Then, normalizing each vector to obtain unit vectors. Here are the steps:

1. Start with the first vector, let's call it V1. This vector will become the first vector in the orthonormal sequence.
2. Take the second vector, V2. Subtract the projection of V2 onto V1 from V2 to obtain a new vector, let's call it u2.
3. Next, take the third vector, V3. Subtract the projection of V3 onto V1 from V3, and subtract the projection of V3 onto u2 from the result, to obtain a new vector, let's call it u3.
4. Continue this process for each remaining vector, subtracting the projections onto all previous vectors in the sequence.
5. Normalize each resulting vector by dividing it by its magnitude to obtain a unit vector.