1. Suppose that U, W are both 5 dimensional subspaces of P7(F). What is the shor

Question

1. Suppose that U, W are both 5 dimensional subspaces of P7(F). What is the shortest possible length for a spanning list of U (intersected with) W?

2. Suppose that W is finite dimensional and T: V(mapped to)W is linear and 1-1. Prove carefully that V is finite dimensional and dimV (less than or equal to) dimW. {Hint: argue by contradiction. Suppose dimW=m and let (v1,...,v(m+1)) be an independent list in V}

3. Prove directly from the definitions: Suppose v1,...,vk,w are elements of a vector space V such that (v1,v2,...,vk) is linearly independent list, but the list (w,v1,...,vk) is linearly dependent. Then w(exists in)span(v1,...,vk).

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