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
(where n is the dimension of the transformation's domain).
Let T:Rn→Rm be a linear
transformation. Then there exists a unique matrix A, called the standard
matrix of T, such that T = TA.
Let A be m x n matrix whose columns are aj =
TA(ej) = Aej = aj
TA = T
If TA(ej) = TB(ej), then A = B.
Consider the linear transformation
T:Rn→Rm and its standard matrix
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
- 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
- rank(A) = m = n
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.