Linear Map Projection

Let $a\in {\mathbb{R}}^2$ be a non-zero vector. It determines a line $L=span(a)\subset {\mathbb{R}}^2$ Consider the linear map $P_n:{\mathbb{R}}^2\to {\mathbb{R}}^2$ given by: $P_a$ where $v\in {\mathbb{R}}^2$ constructing a perpendicular line to $L$ through $v$, this line intersects $L$ at $P_a(v)$

Give a formula for $P_a(v)$ using the inner product $⟨.,.⟩$

Drawing 2025-02-07 11.38.32.excalidraw

⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠

Text Elements

v - ca

Pa(v) = proj_a (v) = <a,v> / <a,a> * a

ca

This orthoganally gives: 0 * <a, > = <a,v> - c<a,a> ^X36GMTxB

Thus,

c = <a,v> / <c,a>

Link to original

This will be $P_a=\frac{⟨a,v⟩}{⟨a,a⟩}\ast a$

Example

Project $(4,3)$ onto $span\{⟨1,1⟩\}$

$P_{⟨1,1⟩}(4,3)=\frac{⟨(1,1),(4,3)⟩}{⟨(1,1),(1,1)⟩}\ast (1,1)$ $=\frac{4+3}{1+1}(1,1)=\frac{7}{2}(1,1)$