Finding linear maps which only act on a subspace

Question

A $n\times n$ matrix (linear map) can act on a $n$-dimensional vector. But say, I want to only find the effect on a $m$-dimensional subspace (in which some of the components of the $n$-dimensional vector lies), where $m<n$. That is, I want to find a $m\times m$ matrix which only acts on a $m$-dimensional subspace containing some of the components of the original $n$-dimensional vector. I know how to project vectors (in this case a $n$-dimensional vector) onto a subspace but don't know how to project linear maps. Any hints/ideas/suggestions?

Answer 1

It's the same method as for any operator $T$ on any finite-dimensional vector space:

1. Find a basis $B = (v_1, \ldots, v_m)$.
2. Compute $Tv_1, Tv_2, \ldots, Tv_m$.
3. For each $i = 1, \ldots, m$, compute the coordinate vector of $Tv_i$ with respect to basis $B$. That is, find the unique scalars $\alpha_{j, i}$ such that $Tv_i = \alpha_{1,i} v_1 + \ldots + \alpha_{m,i} v_m$.
4. Put these scalars into column vectors, then assemble the column vectors into a matrix (all in order). In particular, make the matrix $(M)_{j,i} = \alpha_{j,i}$.

In order to use this matrix, you must first put the vectors from your subspace into coordinate column vectors with respect to $B$. Then multiplying $M$ by this vector results in the coordinate column vector with respect to $B$ of the image of this vector under the transformation.