Problem 3.3. Let T be a linear transformation from V to itself. Define a map T ?
ID: 1948519 • 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
1) (T*f)(ax+by)=f(T(ax+by))=f(aT(x)+bT(y))=af(T(x))+bf(T(y))=a(T*f)(x)+b(T*f)(y) for any a,b in the scalar field, x,y in V and f in V* I have used that both T and f are linear transformations that is T(ax+by)=aT(x)+bT(y) and f(ax+by)=af(x)+bf(y) 2) V* is the space of linear transformations f:V-->F where F is the scalar field( you may have it denoted differently such as k or R ) V* is also vectorial space over F, with the operation a*f=af, where af(x)=a*f(x) for any x (T(af+bg))(x)=(af+bg)(T(x))=(af)(T(x))+(bg)(T(x))=a*f(T(x))+b*g(T(x)) =a*T(f(x))+b*T(g(x)) so T(af+bg)=a*T(f)+b*T(g) for any a,b in F and x in V which shows T is a linear transformation
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