Final answer:
To determine if w is a subspace of v, we check if w is closed under addition and scalar multiplication, contains the zero vector, and is a subset of v. Vector addition is commutative and associative, and scalar multiplication is distributive. A subspace must also include the null vector.
Step-by-step explanation:
When considering if w is a subspace of v, several properties must be satisfied: w must be a subset of v, it must contain the zero vector, it must be closed under vector addition, and it must be closed under scalar multiplication. Simply being a subset does not guarantee that w is a subspace of v. The closure under addition means that if you take any two vectors in w, their sum must also be in w. If taking two steps (or adding two vectors) of different sizes results in ending up at the starting point, this signifies that one vector is the additive inverse of the other, highlighting the property that two vectors of different magnitudes can add to zero.
Addition of vectors is commutative and associative, which ensures that the order of addition does not affect the sum of vectors. Scalar multiplication by a sum of vectors is also distributive, which means that multiplying a scalar by each component of a vector will result in a vector in the same direction, albeit scaled. The null vector, which has a magnitude of zero and contains all components as zero, is essential for a set to be considered a subspace, because the presence of the zero vector among others guarantees the possibility of reaching a 'no displacement' state through vector addition.
Lastly, a change in velocity is indeed a change because velocity is a vector.