radou said:I assume you are speaking of a bounded linear transformation?
If T is bounded, then there exists some constant C so that ||Tx||<=C||x|| for all x from the domain of T, and it follows almost directly from this definition.
CarmineCortez said:Tx >= T(x/||x||)
ystael said:[tex]Tx[/tex] and [tex]T\left(\frac{x}{\|x\|}\right)[/tex] are vectors in the range space of [tex]T[/tex]; they do not possesses an order relation. The relationship between these two vectors is simpler, and follows directly from the definition of a linear transformation.
The norm of a linear transformation is a mathematical concept that measures the 'size' or 'magnitude' of a linear transformation. It can be thought of as the distance between the original vector and the transformed vector.
The norm of a linear transformation is calculated using a specific formula, known as the 'norm formula'. This formula involves taking the square root of the sum of the squares of the components of the transformed vector.
The norm of a linear transformation is important because it helps us understand how much a linear transformation distorts or stretches a vector. It is also used in various mathematical applications, such as optimization and solving linear equations.
No, the norm of a linear transformation is always a positive value. This is because it is calculated using the square root of the sum of squares, which cannot be negative.
The norm of a linear transformation is closely related to the concept of a unit vector. A unit vector is a vector with a norm of 1. When a linear transformation is applied to a unit vector, the resulting vector will have a norm equal to the norm of the transformation. This means that the norm of a linear transformation can be seen as a measure of how much the transformation changes the length of a vector.