Question

Is w a subspace of v? If not, state why. Assume that v has the standard operations.

1) Yes, because w satisfies all the properties of a subspace.
2) No, because w does not satisfy the closure property under addition.
3) No, because w does not contain the zero vector.
4) No, because w does not contain all possible vectors in v.

Answer 1

If w does not adhere to the properties required of a subspace, specifically closure under addition, scalar multiplication, and containing the null vector, it is not a subspace of v. Vector addition is commutative and the null vector acts as the identity element in vector spaces.

Step-by-step explanation:

To determine if w is a subspace of v, several properties must be confirmed:

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If w does not satisfy any one of these properties, it is not a subspace. For instance, if w does not satisfy the closure property under addition (option 2), then it cannot be a subspace. Similarly, if w does not include the zero vector (option 3), it cannot be a subspace as the zero vector acts as the identity element in vector addition.

To address the vector addition principles mentioned, vector addition is indeed commutative; this means A + B = B + A. Moreover, vectors can indeed negate each other when they are equivalent but have opposite directions—this can result in the null vector, which has no length or direction.

To demonstrate the commutative property with three vectors A, B, and C, we can add them in different orders and get the same result. For example, A + B + C and B + C + A will yield the same vector sum.