I am very confused regarding the topic of orthogonal projections, so I will be really thankful if someone could help me.
In my script is written that in order to find the orthogonal projection of a vector onto Space we have to:
find a Basis of the space
orthonormalise it
multiply the vector with the every inner product of the vector and the orthonormlised Basis and we get the orthogonal projection
However, somewhere else there is another formula solving projections like
- from the span we have to make a matrix and find the solution of $A\cdot A^T=A^T\cdot x$ , we get from here a vector
- the vector we got, we multiply with $A$ and that is the orthogonal projection
and also another way solving this like with the Gremian matrix.
So which way is the right, or are all of them right? Which do I have to use in which case and why do we solve it like this?
If you have only one vector $x$, the second method is easier. If you have one space and you want to find a projection of multiple vectors, the first one might be easier because you can precalculate the first two steps.
If you project into a subspace with dimension more than half of the dimension of whole space, it is easier to calculate the rejection to the complement, which is the same.