Gram-Schmidt and Projections

What is the Gram-Schmidt process?

A method to convert a linearly independent set \{v_1,…,v_n\} into an orthonormal basis \{u_1,…,u_n\}: u_1=\frac{v_1}{\|v_1\|}, u_k=\frac{v_k-\sum^{k-1}_{j=1}\langle v_k,u_j\rangle u_j}{\|v_k-\sum^{k-1}_{j=1}\langle v_k,u_j\rangle u_j\|}

What is the orthogonal projection of a vector v onto a subspace W?

The unique vector proj_W(v)\in W such that v-proj_w(v)\in W^{\bot}. It minimizes the distance from v to W.

Projection onto a line spanned by u\in V is calculated by…

proj_u(v)=\frac{\langle v,u\rangle}{\langle u,u \rangle}u

What are common uses for the Gram-Schmidt process?`

The Gram-Schmidt process is used to:

• Construct an orthonormal basis from any linearly independent set

• Simplify computations in QR decomposition

• Make inner product calculations easier

• Facilitate orthogonal projections and applications in least squares problems.