R^3/Kernel of 2x+3y-z/Orthonormal basis/Gram-Schmidt/Example

Let be the kernel of the linear mapping

As a linear subspace of , carries the induced inner product. We want to determine an orthonormal basis of . For this, we consider the basis consisting of the vectors

We have ; therefore,

is the corresponding normed vector. According to orthonormalization process, we set

We have

Therefore,

is the second vector of the orthonormal basis.