Question

What are the conditions of a
linear transformation?

Answer 1

Linear transformations are mathematical functions that preserve the structure of a vector space and satisfy linearity of addition and scalar multiplication.

Step-by-step explanation:

Linear transformations are mathematical functions that map one vector space to another in a way that preserves the structure of the vector space. In order for a transformation to be considered linear, it must satisfy two conditions: linearity of addition and linearity of scalar multiplication.

1. Linearity of addition: This condition states that the transformation of the sum of two vectors should be equal to the sum of the transformed vectors. Mathematically, if T is a linear transformation, it should satisfy T(u + v) = T(u) + T(v), where u and v are vectors.

2. Linearity of scalar multiplication: This condition states that the transformation of a scalar multiple of a vector is equal to the scalar multiple of the transformed vector. Mathematically, if T is a linear transformation, it should satisfy T(ku) = kT(u), where k is a scalar and u is a vector.

Answer 2

Answered

Step-by-step explanation:

A linear transformation T:U→V is a function that carries the elements of the vector space U (Domain) to the vector space V(codomain). Linear Transformation has two additional properties and these are:

`T(u_1+u_2)= T(u_1)+ T(u_2) for\ all\ u\ belonging\ U`

`T(\alpha u)= \alpha T(u) for\ all\ u\ belonging\ C`

These are two defining condition are in the definition of the linear transformation.