The standard matrix of a linear transformation

The standard matrix of a linear transformation is a matrix that induces the transformation. Properties of this matrix will imply properties of the linear transformation itself.

To find the standard matrix of a linear transformation, simply construct a matrix whose columns are the output of the transformation when applied to the standard vectors e₁...e_n (where n is the dimension of the transformation's domain).

Theorem:
Let T:Rⁿ→R^m be a linear transformation. Then there exists a unique matrix A, called the standard matrix of T, such that T = T_A.

Proof:

Existence:
Let A be m x n matrix whose columns are a_j = T(e_j). Then:

T_A(e_j) = Ae_j = a_j
T_A = T

Uniqueness:
If T_A(e_j) = T_B(e_j), then A = B.

Properties:
Consider the linear transformation T:Rⁿ→R^m and its standard matrix A∈M(m,n):

The statements within each of the following groups are equivalent; the groups are not equivalent to each other, nor are statements that exist in separate groups:

• T is one-to-one.
• The equation Ax = b has at most 1 solution for any given b.
• The columns of A are linearly independent.
• rank(A) = n

---

• T is onto.
• The equation Ax = b has at least 1 solution for any given b.
• The columns of A are a spanning set for R^m.
• rank(A) = m

---

• T is invertible
• The equation Ax = b has exactly one solution for any given b.
• The columns of A are linearly independent and span R^m.
• rank(A) = m = n

Example:
Well, MathML doesn't appear to be very widely supported. If it was, there'd be matrices all over the place down here, but, well... Sorry. Feel free to drop me a note if this changes and I don't notice.