5. Consider the following: i- (4 points) If U and V are vector spaces, is their

Question

5. Consider the following: i- (4 points) If U and V are vector spaces, is their intersection Unv a vector space? intersection is the set of all common elements). ii- (4 points) Assume you have a collection of linearly independent vectors v1,..., Un, u1,, um Let V be span of v,...,Un and let U be the span of u,.. m Describe the intersection of U and V and find its dimension. ii- (3 points) Assume now that instead you are given thatv,...,Un,u1,..., Um-1 is a linearly independent collection of vectors, and that ui,..., um is also a linearly independent collection of vectors, and also that thn is in the span of u,..An. As before let V be the span of vi,... , Un and U be the span of u... um. Describe the intersection of U and V, and find its dimension.

Explanation / Answer

Note: If U = Rm and V = Rn, and m n, then UV = is not a vector space as the empty set does not contain   the zero vector.

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