Problem 3.3. Let T be a linear transformation from V to itself. Define a map T ?
ID: 1948333 • Letter: P
Question
Problem 3.3. Let T be a linear transformation from V to itself. Define a map T ? from V ?to itself by the rule
(T?f)(x) = f(Tx), for x ? V,
for all f ? V ?. For this definition, note that since V ? consists of linear functions, the value
of f ? V ? under T ? is a function, so the above rule tells us exactly what the function T ?f
is by showing how it acts on elements of V (compare this with how we defined vector
addition and scalar multiplication on V ?). Using this definition, prove the following:
1. For any f ? V ?, prove that T?f is a linear function, that is, show that T?f ? V ?.
2. Prove that T ? is a linear transformation from V ? to V ? .
The linear transformation T? is called the transpose of the linear transformation.
For those of you familiar with the transpose of a matrix, there is indeed a relationship
between the transpose of a linear transformation and the transpose of a matrix. Next
quarter, you will show that to any linear transformation T, there is an associated matrix
A; then the matrix associated to the linear transformation T? is given by the transpose of
A.
Explanation / Answer
T: V->V is a linear transformation
T*: V*->V* defined as follows:
T*(f) = f(T)
where f(T) is the linear transformation on V that takes x to f(T(x))
To prove: T*(f) is a linear functional
T*(f) (ax+by) where x,y belong to V and a,b are scalars
= f(T(ax+by)) = f(aT(x)+bT(y)) = af(T(x))+bf(T(y)) = aT*f(x) + bT*f(y)
hence T*(f) is infact a linear function with values in the scalar field. ie it's a linear functional
To prove T* is a linear transformation from V* to V*
Consider f and g belonging to V*
T*(af+bg) (x) = (af+bg)(T(x)) = af(T(x))+bg(T(x)) = aT*(f) + bT*(g)
ie, T* acts linearly on V* and hence is a linear transformation on V*.
These lecture notes are well written. "The linear transformation T* is called the transpose of the linear transformation.
For those of you familiar with the transpose of a matrix, there is indeed a relationship
between the transpose of a linear transformation and the transpose of a matrix. "
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.