Orthogonal Projection

Introduction

Orthogonal Decomposition

Orthogonal Projection

Let $W$ be the $xy$-plane in $\mathbb{R}^3$ and let $x = (1,2,3)$. What is the orthogonal projection $x_W$ of $x$ onto $W$?

Let $L$ be the line spanned by $u = (3,2)$ and let $x = (-6,4)$. The projection formula gives $x_L = \dfrac{x\cdot u}{u\cdot u}\,u$. What is the scalar $\dfrac{x\cdot u}{u\cdot u}$?

In two words: writing $x = x_W + x_{W^\perp}$ with $x_W$ in $W$ and $x_{W^\perp}$ in $W^\perp$ is called the orthogonal ______ of $x$ with respect to $W$.

Let $L$ be the line in $\mathbb{R}^2$ spanned by $u = (1,1)$, and let $x = (3,0)$. The projection is $x_L = \dfrac{x\cdot u}{u\cdot u}\,u$. Enter $x_L$ as a comma-separated pair, for example 1,-2.

Let $A$ be the $4\times 3$ matrix $\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \\ 0&0&0 \end{pmatrix}$, whose columns are $e_1,e_2,e_3$ in $\mathbb{R}^4$, and let $b = (1,2,3,4)$. Project $b$ onto $W = \operatorname{Col}(A)$. Enter the four coordinates of the projection, separated by commas, for example 1,0,3,-2.

In your own words, state the single most important definition or theorem of this section, and explain what it says.