Let V be a vector space over a field k, let WCV be any subset, and...

 

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Let V be a vector space over a field k, let WCV be any subset, and let kW = {aw + a2W2 + ... + anwn | ai Ek, w; E W}. Show that kW is a subspace of V. Problem 4. Let k be a field and n E N. Show that if V is an n-dimensional k-vector space, then Vk" as k-vector spaces. Problem 5. Let k C K be a field extension, and let ACK. Show that there exists a smallest subfield K' CK containing k and A. Let V be a vector space over a field k, let WCV be any subset, and let kW = {aw + a2W2 + ... + anwn | ai Ek, w; E W}. Show that kW is a subspace of V. Problem 4. Let k be a field and n E N. Show that if V is an n-dimensional k-vector space, then Vk" as k-vector spaces. Problem 5. Let k C K be a field extension, and let ACK. Show that there exists a smallest subfield K' CK containing k and A.