- #1

bornofflame

- 56

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## Homework Statement

Show that T:C[a,b] -> defined by T(f) = ∫(from a to b) f(x)dx is a linear transformation.

## Homework Equations

Definition of a linear transformation:

A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W, such that

i. T(u+v) = T(u) + T(v) for all u, v in V, and

ii. T(cu) = cT(u) for all u in V and all scalars c

## The Attempt at a Solution

In order for this mapping to be a linear transformation it must follow the above rules, so

i. T(f + g) = ∫ [f(x) + g(x)]dx = ∫ f(x)dx + ∫ g(x)dx = T(f) + T(g)

ii. T(cf) = ∫ cf(x)dx = c∫ f(x)dx = cT(x)

Is it really that simple or did I over simplify the problem?

This problem is not taken from any book (afaik) and doesn't resemble much that we've done in class or from the book so it's throwing me off a bit as I bend my brain to look at things from different angles trying to make sure that I don't miss anything.