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Math Help - linearly independent and dependent

  1. #1
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    linearly independent and dependent

    Problem 1:
    If u_{1}, u_{2},u_{3} are linearly independent vectors in R^{n}, show that \{u_{1}, u_{3} \} is linearly independent set.

    Proof 1:
    If a_{1}u_{1}+a_{3}u_{3}=0 where a_{1}, a_{3} \in R, a_{1}=a_{3}=0 since u_{1},u_{2},u_{3} is linearly independent.

    Is the proof correct? I felt something missing in the proof.



    Problem 2:
    If u_{1}, u_{2},u_{3} are linearly dependent vectors in R^{n}, show that \{u, u_{1}, u_{2}, u_{3} \} is linearly dependent set for any u \in R^{n}.

    Proof 2:
    If au+a_{1}u_{1}+a_{2}u_{2}+a_{3}u_{3}=0. Since u_{1}, u_{2}, u_{3} are linearly dependent, a_{i} \neq 0 for some i=1,2,3. Thus, \{u, u_{1}, u_{2}, u_{3} \} is linearly dependent.

    Please check my proofs.



    Problem 3:
    Let u_{1}, ..., u_{k} \in R^{n} and A \in M_{n}(R), A is invertible .
    Show that u_{1}, ..., u_{k} are linearly independent if and only if u_{1}A, ..., u_{k}A are linearly independent vectors in R^{n}.

    How to prove problem 3? what theorem do I need?
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  2. #2
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    Quote Originally Posted by deniselim17 View Post
    Problem 1:
    If u_{1}, u_{2},u_{3} are linearly independent vectors in R^{n}, show that \{u_{1}, u_{3} \} is linearly independent set.

    Proof 1:
    If a_{1}u_{1}+a_{3}u_{3}=0 where a_{1}, a_{3} \in R, a_{1}=a_{3}=0 since u_{1},u_{2},u_{3} is linearly independent.

    Is the proof correct? I felt something missing in the proof.


    Please check you haven't yet proved anything... . You could argue as follows:

    Suppose 0=a_1u_1+a_3u_3=a_1u_1+0\cdot u_2+a_3u_3\Longrightarrow a_1=0=a_3 since we're given \{u_1,u_2,u_3\} is lin. ind.



    Problem 2:
    If u_{1}, u_{2},u_{3} are linearly dependent vectors in R^{n}, show that \{u, u_{1}, u_{2}, u_{3} \} is linearly dependent set for any u \in R^{n}.

    Proof 2:
    If au+a_{1}u_{1}+a_{2}u_{2}+a_{3}u_{3}=0. Since u_{1}, u_{2}, u_{3} are linearly dependent, a_{i} \neq 0 for some i=1,2,3. Thus, \{u, u_{1}, u_{2}, u_{3} \} is linearly dependent.

    Please check my proofs.


    This is similar as the above one but at least here you mentioned that the given set of vectors is l.i....I'd wrap it up as follows:

    Since \{u_{1}, u_{2},u_{3}\} is l.d. we have a trivial linear combination a_1u_1+a_2u_2+a_3u_3=0 and a_i\neq 0 for some 1\leq i\leq 3\Longrightarrow 0\cdot u+a_1u_1+a_2u_2+a_3u_3=0 and

    STILL a_i\neq 0 for some index i\,\Longrightarrow \{u,u_1,u_2,u_3\} is l.d.



    Problem 3:
    Let u_{1}, ..., u_{k} \in R^{n} and A \in M_{n}(R), A is invertible .
    Show that u_{1}, ..., u_{k} are linearly independent if and only if u_{1}A, ..., u_{k}A are linearly independent vectors in R^{n}.

    How to prove problem 3? what theorem do I need?
    This problem is sensibly more advanced than the other two, which just check you've actually understood the definition of l.d./l.i. sets ; here we need the lemma:

    Lemma: A square matrix n\times n\,\,A is invertible iff \ker A=\{0\}\Longleftrightarrow vA=0\Longleftrightarrow v=0

    Try now to prove the question...

    Tonio
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