## Constructing a projector onto a given subspace

Let be a subspace of Let and fix some basis in Define the matrix of size (the vectors are written as column vectors).

**Exercise 1**. a) With the above notation, the matrix exists. b) The matrix exists. c) is a projector.

**Proof**. a) The determinant of is not zero by linear independence of the basis vectors, so its inverse exists. We also know that and its inverse are symmetric:

(1)

b) To see that exists just count the dimensions.

c) Let's prove that is a projector. (1) allows us to make the proof compact. is idempotent:

is symmetric:

**Exercise 2**. projects onto

**Proof**. First we show that Put

(2)

for any Then

This shows that

Let's prove the opposite inclusion. Any element of is of form with some because we are dealing with a basis. This fact and the general equation imply Hence for any given there exists such that Then (2) is true and, as above, We have proved

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