In the vector space F (−infinity, infinity) any function whose graph passesthrough the origin is a zero vector
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This is false.
Firstly, the vector space F (-infinity, infinity) consists of all real-valued functions that are defined on the interval (-infinity, infinity). A function is considered a vector in this space, and these vectors can be added and scaled just like ordinary vectors in Euclidean space.
Next, a zero vector in this space is a function that, when added to any other function in the space, results in the original function. In other words, it is a function f(x) that satisfies the equation:
f(x) + g(x) = g(x)
for all functions g(x) in the vector space.
Now, consider the function f(x) = 0. This function passes through the origin, since f(0) = 0. However, it is not a zero vector in the vector space because it does not satisfy the equation above. For example, if we choose g(x) = x, then:
f(x) + g(x) = 0 + x = x
which is not equal to g(x).
Therefore, any function that passes through the origin is not necessarily a zero vector in the vector space F (-infinity, infinity).
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