Linear Map Comes from a Matrix: Explanation

Every linear transformation \(L\from \RNrSpc{n}\to \RNrSpc{m}\) may be represented by a matrix, ... which matrix? – Here we explain first how the representing matrix is built. Then we explain which fundamental property of a linear map which makes this result possible.

How to build the matrix representing a linear map   Here we explain the procedure by which we build the matrix \(\Mtrx{A}\) representing a linear transformation \(L\from \RNrSpc{n}\to \RNrSpc{m}\):

1. First apply \(L\) to the vector \(\StdBssVec{1}=(1,0,\dots ,0)\); then use the resulting vector of \(\RNrSpc{m}\) as the first column of \(\Mtrx{A}\).
2. Next apply \(L\) to the vector \(\StdBssVec{2}=(0,1,0,\dots ,0)\), then use the resulting vector of \(\RNrSpc{m}\) as the second column of \(\Mtrx{A}\).
3. etc. until
4. Finally apply \(L\) to the vector \(\StdBssVec{n}=(0,\dots ,0,1)\), then use the resulting vector of \(\RNrSpc{m}\) as the \(n\)-th and last column of \(\Mtrx{A}\).

Why does the representing matrix exist?   Recall that for \(n\geq 1\), \(\RNrSpc{n}\) contains infinitely many points. Therefore the function \(L\) must make infinitely many assignments, one assignment of a point \(L(\Vect{x})\) in \(\RNrSpc{m}\) for each \(\Vect{x}\) in \(\RNrSpc{n}\).

What we learn here is that, because \(L\) is linear, all of these infinitely many assignments are determined by the finite collection of assignments

1. \(L\) sends \(\StdBssVec{1}\) to \(L(\StdBssVec{1})\).
2. \(L\) sends \(\StdBssVec{2}\) to \(L(\StdBssVec{2})\).
3. etc.
4. \(L\) sends \(\StdBssVec{n}\) to \(L(\StdBssVec{n})\).

So this is it: finite base information implies information about infinitely many transformation situations. – What we describe here is a generalization of familiar linear functions like

• Info: A particle moves at the constant speed of \(100km/h\). This is a single bit of information. Yet it enables us to recover the distance traveled by the particle within any interval of time of length \(t\): \(t \cdot 100\)[km].

  In this example, the fact that the particle travels at constant speed means that the distance traveled function \(d\from \RNr\to \RNr\) is a linear function of time. – Without the information that the particle travels at constant speed, we have no way of telling how far it has traveled after 2 hours or 3 hours, etc.

• Info: Henry earns 20 dollars per hour. This is a single bit of information. Yet it enables us to tell how much Henry earns after any number of hours at work. His earnings are a linear function of time. If Henry’s income is by commission only, his earnings are a nonlinear function of time; we have no way of telling how much he will earn after the first two hours if we only know what he earned after one hour.