How can I find the projection of a vector onto a subspace?

1 Answer
Jun 20, 2018

Answer:

See below

Explanation:

Let's say that our subspace #S\subset V# admits #u_1, u_2, ..., u_n# as an orthogonal basis. This means that every vector #u \in S# can be written as a linear combination of the #u_i# vectors:

#u = \sum_{i=1}^n a_iu_i#

Now, assume that you want to project a certain vector #v \in V# onto #S#. Of course, if in particular #v \in S#, then its projection is #v# itself. Let's assume that #v in V# but #v notin S#. Let's call #u# the projection of #v# onto #S#.

Following what we wrote before, we need to find the coefficients #a_i# to express #u# inside #S#. These coefficients are

#a_i = \frac{\langle v, u_i\rangle}{\langle u_i, u_i\rangle}#

So, the final answer is

#u = \sum_{i=1}^n \frac{\langle v, u_i\rangle}{\langle u_i, u_i\rangle}u_i#