True-False Questions! I only need 4 and 5!! Please provide a proof as well thank

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True-False Questions! I only need 4 and 5!! Please provide a proof as well thank you!!

municull tion anything that is not a simple consequence of somell understands nc dl be convinced of the validily 0f True-False Questions Provide a prooffor each true statement and a counterexample for each false 1. A subset of a linearly independent set is linearly independent. 2. A subset of a linearly dependent set is linearly dependent. 3. A set that contains a linearly independent set is linearly independent. 4. A set that contains a linearly dependent set is linearly dependent. 5. If a set of elements of a vector space is linearly dependent, then each element of the set is a linear combination of the other elements of the set. 6. A set of vectors that contains the zero vector is linearly dependent. 7. If X is in the span of A,.A2, and As, then the set (X, A1, A2, A3] is linearly 8. If (X, A,, A2, As) is linearly dependent then X is in the span of A1 A2, and independent as long as the Ai are independent. A3.

Explanation / Answer

4. TRUE. Lets prove it by contradiction. Suppose by contrary that {v1, v2,…,vn} is a linearly independent set, but its subset .{u1, u2,…,um} is linearly dependent.

Since, by "Subset of Linearly Independent Set", a subset {u1, u2,…,um} of {v1, v2,…,vn} must itself be linearly independent. Therefore, set  {v1, v2,…,vn} can not be linearly independent.

Therefore, if {u1, u2,…,um} is linearly dependent, then its superset {v1, v2,…,vn} must be linearly dependent.

5.  If V is a linearly dependent set of a vector space, then each vector is a linear combination of the other vectors in V is FALSE.

We will show it by giving a counter example, Let [2, 1] , [6, 3] and [3, 8] are linearly dependent as vectors [2, 1] , [6, 3] are dependent. But the vector [3, 8] is not a linear combination of vectors [2, 1] , and [6, 3].

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