Vector Projection and Gram-Schmidt Orthogonalization

Given vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^n$: 1. Derive the formula for the projection of $\mathbf{b}$ onto $\mathbf{a}$, and compute the residual (the component of $\mathbf{b}$ orthogonal to $\mathbf{a}$). Verify that the residual is indeed orthogonal to $\mathbf{a}$. 2. Explain how this projection extends to the Gram-Schmidt procedure: given a set of linearly independent vectors $\{\mathbf{v}_1, \ldots, \mathbf{v}_k\}$, produce an orthogonal set $\{\mathbf{u}_1, \ldots, \mathbf{u}_k\}$ spanning the same subspace. 3. How does Gram-Schmidt relate to the QR decomposition?