Shortest distances in $\R^2$

The nearest point from $B$ on the line passing through $A$ and the origin is $v= \frac {\langle a,b \rangle} {\langle a,a \rangle} a$.

The Projection of a Vector to a Subspace

The projection of $v$ onto $W$ is the vector in $W$ closest to $v.$

Projection on a Vector and Orthogonal Bases.

Let $V$ be an inner product space and $v,w \in V$

→ Define $proj_w(v)=proj_{\langle w \rangle} (v)$ , here $w$ is a vector and not a subspace.

→ An orthonormal basis for $\langle w \rangle$ is $\frac w {||w||}$.

Projection as a Linear Transformation.

Let $V$ be an inner product space and $W$ be a subspace.

Then the projection of vectors in $V$ to $W$ is a linear transformation($P_W$) from $V$ to $V$ with image $W$.