In Exercise 72 through 74 , let be the set of all polynomials of degree such that f(0) = 0.
73. Is the linear transformation an isomorphism from to ?
The linear transformation is an isomorphism from to .
Consider two linear spaces and . A function from to is called linear transformation if
for all elements f and g of V and for all scalars .
If V is finite dimensional, then
An invertible linear transformation T is called an isomorphism.
Consider the linear transformation given by .
Consider any non-zero polynomial,
On applying the limits,
Since, it non-zero there exists localid="1659421702278" such that .
Thus, is also a non-zero polynomial in .
Thus, T is an isomorphism.
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