Given the basis for an orthogonal complement, how would you find the matrix that projects all vectors onto this basis?

Question

Let's say $A = \text{projection onto orthogonal complement}, V^{\perp}$. Are the columns of $A$ just made up of the basis vectors of $V^{\perp}$ or do we have to do some extra work (and in which case, what must be done)? We just covered projections in class and I'm slightly confused by how they work, so any input is appreciated!

Answer 1

If your vector space $U$ is finite-dimensional and $V$ is a subspace, then it decomposes as $U=V\oplus V^\perp$. This means that giving a vector in $U$ is the same thing as giving a vector in $V$ and a vector in $V^\perp$. Consequently, defining a linear transformation in $U$ is really the same thing as defining how it acts on $V$ and how it acts on $V^\perp$.

The transformation $A$ that you define as projection onto the subspace $V^\perp$ is then clear from this point of view: it acts as the identity in $V^\perp$ and trivially on $V$ (i.e. maps everything to $0$).

So, if you have a basis of $n$ vectors of $U$ that is made up of a basis of $n-k$ vectors of $V$ and a basis of $k$ vectors of $V^\perp$, the matrix representation of $A$ is just \begin{bmatrix} 0 & 0\\ 0 & I \end{bmatrix}

where $I$ is the $k\times k$ identity matrix.