Question

State whether true or false
Any one-one linear map from C[0, 1] to itself is also onto.

Answer 1

Concept-

"One- One" and "Onto" are properties of functions in general, not just linear transformation.

Given-

Any linear map from C[0,1] is one-one.

Find-

State whether true or false, if any one-one linear map from C[0,1] to itself is also onto.

Solution-

Definition- Let f:X→Y be a function.

• f is one-one if and only if for every y∈ Y there is at most one x∈ X such that f(x) = y ; equivalently , if and only if f(x1) = f(x2) implies x1=x2.
• f is onto ( or onto Y , if the codomain is not clear from context) if and only if for every y∈ Y there is at least one x∈ X such that f(x) = y.

This definition applies to linear transformation as well, and in particular for linear transformations T: R ^n → R ^m , and by extension to matrices, since an m × n matrix A can be identified with the linear L(x) =Ax.

So, the definitions are for any functions. But when our sets X and Y have more structure to them, and the functions are not arbitrary, but special kinds of functions, we can obtain other ways of characterizing a function as one-one to onto which is easier/better/more useful/more conceptual/has interesting applications. This is indeed the case when we have such a rich structure as linear transformations and vector spaces.

One-to-one is probably the easiest ; this is because whether a function is one-to-one depends only on its domain , and not on its codomain. By contrast, whether a function is onto depends on both on the domain and the codomain ( so, for instance, f(x)=x² is onto if we think of it as a function f: R → [0,∝] , but not if we think of it as a function

f: R → R, or f: [2,∝) → [0,∝)).

Hence, it is true that if any one-one linear map from C[0,1] to itself is also onto.

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