linear projection

let $\text{[math]}$ be a vector space and $\text{[math]}$ a subspace of $\text{[math]}$ . a linear mapping $\text{[math]}$ is called a projection if $\text{[math]}$ .
[cite:;taken from @ml_faisal_2020 definition 3.10]

a projection essentially transforms points in $\text{[math]}$ -space to $\text{[math]}$ -space, such that $\text{[math]}$ .

let $\text{[math]}$ be a vector space and $\text{[math]}$ and $\text{[math]}$ be subspaces of $\text{[math]}$ such that $\text{[math]}$ . the function $\text{[math]}$ defined by $\text{[math]}$ where $\text{[math]}$ with $\text{[math]}$ and $\text{[math]}$ , is called the projection of $\text{[math]}$ on $\text{[math]}$ or the projection on $\text{[math]}$ along $\text{[math]}$ .
[cite:;taken from @algebra_insel_2019 chapter 2 linear transformations and matrices]

another definition from a lecture:

let $\text{[math]}$ be a vector space. a linear operator $\text{[math]}$ is called a projection operator (or simply a projection) if $\text{[math]}$ . equivalently, for every $\text{[math]}$ , $\text{[math]}$ .

$\text{[math]}$ .
given two subspaces $\text{[math]}$ such that $\text{[math]}$ , one can define a projection $\text{[math]}$ such that $\text{[math]}$ and $\text{[math]}$ . (this is projection onto $\text{[math]}$ along $\text{[math]}$ ).
$\text{[math]}$ and $\text{[math]}$ are uniquely determined by $\text{[math]}$ . (the subspaces $\text{[math]}$ and $\text{[math]}$ onto which and along which projection occurs are unique).

the formula for projecting a vector $\text{[math]}$ onto the line spanned by another vector $\text{[math]}$ is:
$\text{[math]}$

let $\text{[math]}$ be a finite-dimensional subspace of an inner product space $\text{[math]}$ , and let $\text{[math]}$ . then there exist unique vectors $\text{[math]}$ and $\text{[math]}$ such that $\text{[math]}$ . furthermore, if $\text{[math]}$ is an orthonormal basis for $\text{[math]}$ , then
$\text{[math]}$ [cite:;taken2 from @algebra_insel_2019 chapter 6.2 gram-schmidt orthogonalization process]