Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, from to .
The transformation is a linear transformation and is an isomorphism.
Consider two linear spaces V and W . A transformation T is said to be a linear transformation if it satisfies the properties,
For all elements f,g of v and is scalar.
An invertible linear transformation is called an isomorphism.
Consider the transformation , from to .
Check whether the transformation satisfies the below two conditions or not.
Verify the first condition.
Let and be arbitrary complex numbers from . Then,
It is clear that, the first condition is satisfied.
Verify the second condition.
Let k be an arbitrary scalar, and as follows.
It is clear that, the second condition is also satisfied.
Thus,T is a linear transformation.
A linear transformation is said to be an isomorphism if and only if and .
Now, check whether ker (T) .
According to the definition of the kernel of a transformation,
Consider a complex number A as A = x + iy
Comparing both sides, it can be concluded that x = 0 , y = 0
Now, check whether Im (T) .
According to the definition of the image of a transformation,
Let is in .
, say (B = - Y + ix).
T (A) = B
It is clear that, .
This means for any localid="1659422430378" there exists a localid="1659422483414" such that .
So, Im (T) =
Therefore, the transformation T is an isomorphism.
Thus, the transformation T is a linear transformation and T is an isomorphism.
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