Thank you for your help! 32. If f : X Y and if U X, then we can create a new fun

Question

Thank you for your help! 32. If f : X Y and if U X, then we can create a new function g by the equa- tion g(x) = f(x) for all s in U. If U is a proper subset of X, then g + f because they have different domains. The function g is called the restriction of f to U. The no- tation g = fU is often used. Suppose now that X and Y are linear spaces and L is a lin ear map from X to Y.Let U be a subspace of x. Is LIU linear? What is its domain? Do L and LIU have the same range? (Arguments and examples are needed.)

Explanation / Answer

For linear map L, following conditions must be specified-

1) L(x+y) = L(x) + L(y)

2) L(kx) = kL(x) where k is a scalar.

L : X ------> Y is a linear map. So for every element x in X must satisfy above two conditions. L|U (suppose call it M) will be defined as-

M : U -------> Y where U is proper subset of X and M(x) = L(x) for every x in U. Now for L|U (M) to be linear, conditions 1) and 2) must be satisfied for every element of X. Suppose u is an element in U then u will also be an element of X (U is subset of X). similarly take another element v which will be an element of X. Since u and v are elements of X. so -

M(u+v) = M(u) + M(v) {beacause M(u) = L(u) and these condition are satisfied by L beacause L is linear}

M(ku) = kM(u) and M(kv) = kM(v)

These conditions will be satisfied for every element of U (every element will also be an element of X). Hence M satisfises the conditions required for linear map. Hence M or L|U is linear.

Domain of L|U will be U beacause L|U is defined for elements of U.

Range of L|U and L will not be same beacause L is defined for elements which are in X but not U while L|U is not defined for elements outside U. For such elements, value of mapping is obtained in L but not in L|U.

Example-

L : R ------> R L(x) = x

L|U : R+ --------> R L|U(x) = x (defined for only positive real number)

L and L|U both satisfises the necessary condition for linear map (condition 1) and condition 2)).

Domain of L|U is R+.

Range of L is R but range of L|U is R+ beacause L|U is defined as L|U(x) = x for only positive real number. So this mapping maps positive real numbers to positive real numbers. Hence range of L is not same as L|U.

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