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Any linear transformation can be written as..
A matrix equation
Theorem 10: Let T: R^n ➡R^m be a linear transformation. Then there exist…
A unique matrix A such that T(x) = Ax for all x in R^n
What is the difference between linear transformation and matrix transformation?
The term linear transformation focuses on a property of a mapping, while matrix transformation describes how such mapping is implemented
Reflection through the x1 axis
Reflection through the x2 axis
Reflection through the line x2 = x1
Reflection through the line x2 = -x1
Reflection through the origin
Horizontal contraction and expansion
Vertical contraction and expansion
Horizontal shear
Vertical shear
Onto Transformation
A transformation T: R^n ➡R^m is said to be onto R^n if each b in R^m is the image at least one x in R^n
How do we know if the transformation is onto?
Theorem 4
Ax=b has a solution for all b in R^m
Each b is a linear combination of the columns of A
The columns of A span R^m
Matrix A has a pivot position in every row
One to one transformation
A transformation T: R^n ➡R^m is one to one if each vector b in R^m is the image of at most 1 vector x in R^n
How do we know if the transformation is one to one?
The solution has to be a unique solution
No free variables
Pivot positions in every column
The equation T(x) = 0 has only the trivial solution
The columns of A are linearly independent