(1 point) Are the following statements true or false? 1. If the set of vectors U
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Question
(1 point) Are the following statements true or false? 1. If the set of vectors U is linearly independent in a subspace S, then vectors can be added to to create a basis for S 2. If S, and S2 are subspaces of Rn of the same dimension, then S 3. If the set of vectors U is linearly independent in a subspace , then vectors can be removed from U to create a basis for 4. If the set of vectors U spans a subspace S, then vectors can be removed from U to create a basis for S 5. If S = span(ui, u2, u3}, then dim(S) = 3 True False ,-S2. True False TrueExplanation / Answer
Answer:
1)True.By the theorem that says let U={u1,.......um} be set of vectors in a subspace S!={0} of Rn.If U is linearly independent,then additional vectors can be added to U to form basis for S.
2)False.By the theorem if S1 and S2 are subspaces of Rn of the same dimension then S1 is not equal with S2 .
3)True.By the theorem that says let U={u1,.......um} be set of vectors in a subspace S!={0} of Rn.If U is linearly independent then vectors can be removed from U to form basis for S.
4)True.By the theorem that says let U={u1,.......um} be set of vectors in a subspace S!={0} of Rn.If U spans S then vectors can be removed from U to form basis for S.
5)False.If S={u1,u2,u3} then dim(span(S))=3.
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