Standard matrices

A linear transformation is described completely once you describe how it transforms the elementary vectors of its domain.

If T:Rn → Rm is a linear transformation for which the vectors T(e1), T(e2), ... , T(en) are all known vectors in Rm, then T(u) is known for every vector u in Rn.

Proof: Suppose u = (u1, u2, ..., un) is a typical vector in Rn. Write u as a linear combination of the elementary vectors in Rn:

u = u1e1 + u2e2 + ... + unen.

Then since T preserves linear combinations,

T(u) = u1T(e1) + u2T(e2) + ... + unT(en).

Since T(e1), T(e2), ... , T(en) are all known, this gives us a formula for T(u) for any u in Rn.

 

It turns out that if we represent vectors by column matrices, then linear transformations and matrix transformations are the same thing.

a) Given an m x n matrix A, define the matrix transformation T:Rn → Rm by T(u) = Au for all u in Rn. Then T is a linear transformation and the columns of A are T(e1), T(e2), ... , T(en).

b) If T:Rn → Rm is a linear transformation and A is the matrix with columns T(e1), T(e2), ... , T(en), then T is the matrix transformation with matrix A, i.e. T(u) = Au for all u in Rn.

Proof of a): Suppose T is a matrix transformation with matrix A, i.e. suppose T(u) = Au for all u in Rn. Then

T preserves sums: for vectors u and v in Rn,
         T(u + v) = A(u + v) = Au + Av = T(u) + T(v).

T preserves scalar multiples: for any vector u in Rn and any scalar c,
         T(cu) = A(cu) = cAu = cT(u).

Then T is a linear transformation. Also, T(ek) = Aek, which is the kth column of A.

Proof of b): For any u in Rn, T(u) = u1T(e1) + u2T(e2) + ... + unT(en). The right-hand side of this relation is a linear combination of columns. Use the multiplication pattern you learned in Section 1B to write it in contracted form: if A is the matrix with columns T(e1), T(e2), ... , T(en), then T(u) = Au.

 

The matrix A with columns T(e1), T(e2), ... , T(en) is called the standard matrix for T. The standard matrix of the identity transformation on Rn is the n x n identity matrix. The standard matrix of the zero transformation from Rn to Rm is the m x n zero matrix.

 

To find the standard matrix of a linear transformation

If T:Rn → Rm is a linear transformation

Calculate T(e1), T(e2), ... , T(en).

Form the matrix A with columns T(e1), T(e2), ... , T(en).

Then T(u) = Au for all u in Rn.