Question

Let w be the set of all functions such that f(6) = 10 . is this a subspace of f[a, b] where we have a ≤ 6 ≤ b ?

Answer 1

The set w, consisting of functions where f(6) = 10, is not a subspace of f[a, b] because it does not contain the zero function, thus violating one of the subspace criteria.

Step-by-step explanation:

You asked whether the set w, consisting of all functions such that f(6) = 10, is a subspace of f[a, b] where a ≤ 6 ≤ b. To determine if w is a subspace, we must check if it satisfies the three subspace criteria:

1. It must contain the zero function (the function f(x) such that for all x in the domain, f(x) = 0),
2. It must be closed under vector addition (if f and g are in w, then f + g must also be in w),
3. It must be closed under scalar multiplication (if f is in w and c is any scalar, then cf must be in w).

In the context of f[a, b], the zero function is the function that assigns the value 0 to all x in [a, b]. However, the set w does not include this function because it requires that f(6) = 10, which violates the first criterion. Therefore, w is not a subspace of f[a, b] because it does not contain the zero function.