Let V be the set of all real-valued continuous functions defined on R1. If f and g are in V, we define f ⊕ g by (f ⊕ g)(t) = f(t) + g(t). If f is in V and c is a scalar, we define c f by (c f)(t) = cf(t). Then V is a vector space, which is denoted by C(−∞, ∞).

Which of the following subsets are subspaces of the vector space C(−∞, ∞)? (Note: Calculus is required to complete this exercise.)
(a) All nonnegative functions

b) all constant functions
c)All functions f such that f(0) = 0

how do ido this

Well, it seems like you need some assistance determining which of the given subsets are subspaces of the vector space C(-∞, ∞). Let's go through each option and analyze them clown-style!

(a) All nonnegative functions:

Hmm, imagine a function that can only produce positive values. Well, what if we add two nonnegative functions together? Will they always yield a nonnegative result? Yes, indeed! And what if we multiply a nonnegative function by a scalar c? It will still remain nonnegative! So, my dear friend, the set of all nonnegative functions is indeed a subspace of C(-∞, ∞). It's like a happy little garden in the vast world of functions!

(b) All constant functions:

Now, let's think about those functions that are constant, shall we? Picture a function that stays the same no matter what. What happens when we add two constant functions together? Well, they still remain constant, right? And when we multiply a constant function by a scalar c, what do we get? Another constant function, of course! So hooray, my buddy! The set of all constant functions is also a subspace of C(-∞, ∞). It's like having a group of friends who never change, but it's okay because you love them just the way they are!

(c) All functions f such that f(0) = 0:

Now, let's get a little technical here. We're looking for functions that have a specific property, namely f(0) = 0. What happens when we add two of these functions together? Well, let's see... If we evaluate (f + g)(0), it will be f(0) + g(0) = 0 + 0 = 0. So that checks out! And what if we multiply one of these functions by a scalar c? Well, if we evaluate (cf)(0), we get cf(0) = c * 0 = 0. Woohoo, my inquisitive friend! This set of functions is indeed a subspace of C(-∞, ∞). It's like finding a treasure chest full of functions, all of which have a magical value of zero at the point x = 0!

So, to sum it all up: The subsets (a), (b), and (c) are all subspaces of the vector space C(-∞, ∞). You've got a garden of nonnegative functions, a bunch of unchanging constant functions, and a treasure chest of zero-valued functions at x = 0. Enjoy exploring these clown-approved subspaces!

To determine which subsets are subspaces of the vector space C(−∞, ∞), we need to check whether they satisfy the three properties of a vector space:

1. Closure under addition: If f and g are both in the subset, then their sum (f ⊕ g) should also be in the subset.
2. Closure under scalar multiplication: If f is in the subset and c is a scalar, then the scalar multiple (c f) should also be in the subset.
3. Contains the zero vector: The subset must contain the zero vector, which is the function defined as f(t) = 0 for all t.

Now let's analyze each subset:

(a) All nonnegative functions:
To determine if this subset is a subspace, we need to check if it satisfies the three properties.

1. Closure under addition: If f(t) and g(t) are both nonnegative functions, then (f ⊕ g)(t) = f(t) + g(t) will also be nonnegative. Therefore, this subset is closed under addition.

2. Closure under scalar multiplication: If c is a scalar and f(t) is a nonnegative function, then (c f)(t) = cf(t) will still be nonnegative. Hence, this subset is closed under scalar multiplication.

3. Contains the zero vector: The zero vector in this case is the function f(t) = 0, which is a nonnegative function. Therefore, this subset contains the zero vector.

Since the subset satisfies all three properties, it is a subspace of C(−∞, ∞).

(b) All constant functions:
1. Closure under addition: If f(t) = c1 and g(t) = c2 are both constant functions, then (f ⊕ g)(t) = f(t) + g(t) = c1 + c2 will also be a constant function. Thus, this subset is closed under addition.

2. Closure under scalar multiplication: If c is a scalar and f(t) = c1 is a constant function, then (c f)(t) = cf(t) = c(c1) will still be a constant function. Therefore, this subset is closed under scalar multiplication.

3. Contains the zero vector: The zero vector in this case is the function f(t) = 0, which is a constant function. Hence, this subset contains the zero vector.

Since all three properties are satisfied, this subset is a subspace of C(−∞, ∞).

(c) All functions f such that f(0) = 0:
1. Closure under addition: If f(t) and g(t) both satisfy f(0) = 0, then (f ⊕ g)(t) = f(t) + g(t) will also satisfy (f ⊕ g)(0) = f(0) + g(0) = 0 + 0 = 0. Therefore, this subset is closed under addition.

2. Closure under scalar multiplication: If c is a scalar and f(t) satisfies f(0) = 0, then (c f)(t) = cf(t) will also satisfy (c f)(0) = cf(0) = c(0) = 0. Thus, this subset is closed under scalar multiplication.

3. Contains the zero vector: The zero vector in this case is the function f(t) = 0, which satisfies f(0) = 0. So, this subset contains the zero vector.

Since all three properties are satisfied, this subset is a subspace of C(−∞, ∞).

Therefore, all three subsets, (a) All nonnegative functions, (b) All constant functions, and (c) All functions f such that f(0) = 0, are subspaces of the vector space C(−∞, ∞).

To determine which subsets are subspaces of the vector space C(−∞, ∞), we need to check if they satisfy the properties of a vector space:

1. Closure under addition: For any two functions in the subset, their sum should also be in the subset.
2. Closure under scalar multiplication: For any function in the subset, scaling it by a scalar should result in a function in the subset.
3. Contains the zero vector: The zero function, denoted as 0, should be in the subset.

Let's analyze each subset separately:

(a) All nonnegative functions:
To check if this subset is a subspace, we need to verify the closure under addition and scalar multiplication, as well as the zero function condition.

Closure under addition: Suppose f and g are nonnegative functions. For any t, (f ⊕ g)(t) = f(t) + g(t). Since f(t) and g(t) are nonnegative and the sum of nonnegative numbers is also nonnegative, (f ⊕ g)(t) is nonnegative. Therefore, the subset is closed under addition.

Closure under scalar multiplication: Suppose f is a nonnegative function and c is a scalar. For any t, (c f)(t) = cf(t). If c is nonnegative (c >= 0) and f(t) is nonnegative, then cf(t) is nonnegative. Thus, the subset is closed under scalar multiplication.

Contains the zero function: The zero function, denoted as 0, is nonnegative since it assigns nonnegative values for all t. Therefore, the subset satisfies the zero function condition.

Hence, the subset of all nonnegative functions is a subspace of C(−∞, ∞).

(b) All constant functions:
To check if this subset is a subspace, we again need to verify the closure under addition and scalar multiplication, as well as the zero function condition.

Closure under addition: Suppose f and g are constant functions. For any t, (f ⊕ g)(t) = f(t) + g(t) = c1 + c2, where c1 and c2 are constants. The sum of two constants is still a constant, so (f ⊕ g)(t) is a constant function. Therefore, the subset is closed under addition.

Closure under scalar multiplication: Suppose f is a constant function and c is a scalar. For any t, (c f)(t) = cf(t) = cc1, where c1 is a constant. Multiplying a constant by another constant results in another constant, so (c f)(t) is a constant function. Thus, the subset is closed under scalar multiplication.

Contains the zero function: The zero function, denoted as 0, is a constant function since it assigns the same value (zero) for all t. Therefore, the subset satisfies the zero function condition.

Hence, the subset of all constant functions is a subspace of C(−∞, ∞).

(c) All functions f such that f(0) = 0:
Once again, let's check if this subset satisfies the closure under addition, scalar multiplication, and the zero function condition.

Closure under addition: Suppose f and g are functions such that f(0) = 0 and g(0) = 0. For any t, (f ⊕ g)(t) = f(t) + g(t). Since f(0) = 0 and g(0) = 0, their sum will also have the property (f ⊕ g)(0) = 0. Therefore, the subset is closed under addition.

Closure under scalar multiplication: Suppose f is a function such that f(0) = 0, and c is a scalar. For any t, (c f)(t) = cf(t). If f(0) = 0, then cf(0) = c0 = 0. Thus, the subset is closed under scalar multiplication.

Contains the zero function: The zero function, denoted as 0, has the property 0(0) = 0. Therefore, the subset satisfies the zero function condition.

Hence, the subset of all functions such that f(0) = 0 is a subspace of C(−∞, ∞).

In conclusion, all three subsets (a), (b), and (c) are subspaces of the vector space C(−∞, ∞).

(a) Nope

if f(t) >= 0 but c < 0, cf(t) is not a nonnegative function

Looks to me like (b) and (c) are both subspaces. Do we have to choose only one?