Yes, you are. It makes no sense to talk about a transformation as being "closed under addition and scale factorization". Only sets are "closed" under given operations, not transformations. A "linear transformation" on a vector space is a function, F(v), such that F(au+ bv)= aF(u)+ bF(v). They are certainly NOT, in general, "one-to-one", as Office Shredder says. For example, any projection is linear but is not one-to-one.yaganon said:right now, my concept for their difference is that linear transformations are 1 to 1, where as nonlinear transformations are not. However, P^n to P^(-1) is a linear transformation, but it's not 1 to 1.
the textbook def of linear transformation is that it must be closed under addition and scale factorization.
I'm confused.
A linear transformation is a mathematical function that maps a vector space to another vector space in a way that preserves addition and scalar multiplication. This means that the output of the transformation is directly proportional to the input, and there is no bending or curving in the transformation.
A nonlinear transformation is a mathematical function that does not preserve addition and scalar multiplication. This means that the output of the transformation is not directly proportional to the input, and there is bending or curving involved in the transformation.
Some examples of linear transformations include translations, rotations, reflections, and dilations. In mathematics, linear transformations are often represented by matrices, and the most common linear transformations are those that involve multiplication by a constant or addition of a constant.
Some examples of nonlinear transformations include polynomial functions, trigonometric functions, logarithmic functions, and exponential functions. These functions involve operations such as squaring, taking the sine or cosine, or raising to a power other than 1, which results in a nonlinear transformation.
The main difference between linear and nonlinear transformations is that linear transformations preserve addition and scalar multiplication, while nonlinear transformations do not. This means that linear transformations result in straight lines or planes, while nonlinear transformations result in curves or surfaces. Additionally, linear transformations can be represented by matrices, while nonlinear transformations cannot be represented by a single matrix.