1.8 The Matrix of a Linear Transformation
- Standard matrix
- Therem8
Let T: R^n -> R^m be a linear transformation. Then there exists a unique matrix A such that
T(x) = Ax for all x in R^n
In fact, A is the m x n matrix whose j-th column is the vector T(ej),
where ej is the j-th column of the identity maxtirx in R^n:
A= [T(e1) ... T(en)]
x= lnx = [e1 ... en]x
= x1e1 + ... + xnen
T(x) = Bx
linear transformation
T(ej) = Bej = bj
= [T(e1) ... T(en)]
R^n에서 R^m으로 transformation 된 결과는 1개이다
그것을 standard matrix라고 한다.
- Geometric Linear Transformation of R^2
- Onto
A mapping T: R^n -> R^m is said to be onto R^m if each b in R^m is the image of at least one x in R^n
Does T map R^n onto R^m?
=> Does T(x) = b have at leat one solution for each b in R^m?
R^n space에서 R^m으로 보낼 때 T(x) = b 가 되는 해가 최소한 1개 이상 있으면 onto라고 표현한다.
- One-to-one
A mapping T: R^n -> R^m is said to be one to one if each b in R^m is the image of at most on x in R^n
Is T one-to-one?
Does T(x) = b have either a unique solution or none at all?
- Theorem11
Let T : R be a linear transformation
Then T is one-to-one if and only if the equation T(x) = 0
has only trivial solution.
T(0) = T(00) = 0T(0) = 0
One-to-One 일대일 대응이며 trivial solution이 있을 때에만 해당한다.
- Theorem12
Let T : R be a linear transformation and let A be
the standard matrix for T. Then:
T maps R^n onto R^m if and only if the columns of A span R^m
T is one-to-one if and only if the columns of A are linearly independent
두 벡터 [[3], [5], [1]] 과 [[1],[7],[3]]가 서로 multiple로 표현될 수 없다.
그러므로 서로 independent이다
independent이면 trivial solution
즉 linear transforamtion에서는 one-to-one이다.
