Let V be a vector space over the field F and let X 1 and X 2 be two non-empty su

Question

Let V be a vector space over the field F and let X1 and X2 be two non-empty subsets of V. Define the new sets:

Explanation / Answer

To prove a set to be subspace,it is sufficient to show that it is closed under vector addition and scalar multiplication Given U and W are subspaces (1) To prove U intersection W as subspace Let v1,v2 belong to U intersection W => v1 belongs to U and v2 belongs to U v1 belongs to W and v2 belongs to W As U and W are subspaces v1+v2 belongs to U as well as W => v1+v2 belongs to U intersection W => U intersection W is closed under vector addition Let p belongs to F and v belongs to U intersection W => v belongs to U and v belongs to W As U and W are subspaces pv belongs to U as well as W => pv belongs to U intersection W => U intersection W is closed under scalar multiplication Hence U intersection W is a subspace (2) Let u1+w1,u2+w2 belong to U+W => u1,u2 belong to U and w1,w2 belong to W => u1+u2 belong to U and w1+w2 belong to W (as U and W are subspaces) => (u1+w1)+(u2+w2) belongs to U+W => U+W is closed under vector addition Let p belongs to F and u+w belongs to U+W p(u+w) = pu+pw Now pu belongs to U and pw belongs to W (as U and W are subspaces) => p(u+w) belongs to U+W => U+W is closed under scalar multiplication Hence U+W is a subspace

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.