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lkh1986
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Algebra- Vector space and subspace
Here are some true or false statements given in my test.
(a) R^2 is a subspace of R^3.
(b) If {v1, v2, ..., vn} is a set of linearly dependent vectors, then it contains a zero vector.
(c) If {v1, v2, ..., vn} is a spanning set, then {v1, v2, ..., vn} are linearly independent.
(a) True, because R is a subspace of R^2 and R^2 is a subspace of R^3 and R^3 is a subspace of R^4, and so on.
(b) False, because it may or may not contain a zero vector. I think that it is true for this statement: If {v1, v2, ..., vn} contains a zero vector, then it is linearly dependent. But the statement "If {v1, v2, ..., vn} is a set of linearly dependent vectors, then it contains a zero vector." is false.
(c) False. Beacuse vectors in spanning sets can be expressed as linear combinations of each others, and hence it is consistent and they are linearly dependent.
Any opinion on these questions? Thanks.
Homework Statement
Here are some true or false statements given in my test.
(a) R^2 is a subspace of R^3.
(b) If {v1, v2, ..., vn} is a set of linearly dependent vectors, then it contains a zero vector.
(c) If {v1, v2, ..., vn} is a spanning set, then {v1, v2, ..., vn} are linearly independent.
Homework Equations
The Attempt at a Solution
(a) True, because R is a subspace of R^2 and R^2 is a subspace of R^3 and R^3 is a subspace of R^4, and so on.
(b) False, because it may or may not contain a zero vector. I think that it is true for this statement: If {v1, v2, ..., vn} contains a zero vector, then it is linearly dependent. But the statement "If {v1, v2, ..., vn} is a set of linearly dependent vectors, then it contains a zero vector." is false.
(c) False. Beacuse vectors in spanning sets can be expressed as linear combinations of each others, and hence it is consistent and they are linearly dependent.
Any opinion on these questions? Thanks.
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