Projection and the Gram-Schmidt Process

The goal this class is to find orthonormal basis for a subspace.

Example:

• What is an orthonormal basis for the span of {(2, 0), (1, 1)}?
• How would we do it with 3 vectors?

Definition: Let u, v ∈ ℝⁿ with v nonzero. Then the projection of u onto v is given by $\mathrm{proj}_v u=\frac{u\cdot v}{\|v\|^2} v$.

Theorem: Let u, v ∈ ℝⁿ and c be a nonzero scalar. Then

• proj_vu is in spanv.
• u − proj_vu is orthogonal to v
• if u is in spanv, then u = proj_vu.
• proj_vu = proj_cvu.

Let S be a nontrivial subspace with orthogonal basis {v₁, …, v_k}. Then the projection of u onto S is given by $\mathrm{proj}_S u=\sum\sb{i=1} ^k \mathrm{proj}\sb{v_i} u$.

Theorem: Let S be a nonzero subspace of ℝⁿ with orthogonal basis {v₁, …, v_k}, and let u be a vector in ℝⁿ. Then

• proj_Su is in S.
• u − proj_Su is orthogonal to S.
• if u is in S, then u = proj_Su.
• proj_Su is independent of the choice of orthogonal basis for S.

Theorem: (The Gram-Schmidt Process) Let S be a subspace with basis {s₁, …, s_n}. Define v₁, …, v_k by

• v₁ = s₁,
• $v_2=s_2-\mathrm{proj}\sb{v_1}s_2$,
• $v_3=s_3-\mathrm{proj}\sb{v_1}s_3-\mathrm{proj}\sb{v_2}s_3$,
• …
• $v_k=s_k-\mathrm{proj}\sb{v_1}s_k-\ldots-\mathrm{proj}\sb{v\sb{k-1}}s_k$

Then {v₁, …, v_k} is an orthogonal basis. To make it orthonormal, just normalize each element.

Example: Find an orthonormal basis for the subspace (1, 0, 1, 1), (0, 2, 0, 3), ( − 3,  − 1, 1, 5).