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...en (where n is the dimension of the transformation's domain).

Theorem:
Let T:RnRm be a linear transformation. Then there exists a unique matrix A, called the standard matrix of T, such that T = TA.

Existence:
Let A be m x n matrix whose columns are aj = T(ej). Then:
TA(ej) = Aej = aj
TA = T

Uniqueness:
If TA(ej) = TB(ej), then A = B.

Properties:
Consider the linear transformation T:RnRm 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 onto.
• The equation Ax = b has at least 1 solution for any given b.
• The columns of A are a spanning set for Rm.
• 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 Rm.
• 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.

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