Let A and B be n by n matrices . Prove that if A is symmetric and B be skew-symmetric , then {A,B} is a linearly independent set.

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As you will encounter in most scenarios when trying to prove that a set is linearly independent, first assume the set is linearly dependent. That is, assume there exists $\alpha,\beta\in\mathbb{R}$ (I assume you're working with matrices with entries in $\mathbb{R}$, but this can be generalized) such that $$ \alpha A +\beta B=0. $$ Then we apply the symmetries of the matrices. Since $A$ is symmetric, and $B$ is skew-symmetric, we have $$ \alpha A -\beta B=\alpha A^T+\beta B^T=0. $$ Then, we have the following two equations: $$ \alpha A=-\beta B, \qquad \alpha A=\beta B. $$ Since $A$ and $B$ are nonzero, we must have that $\alpha=\beta=0$ (a little work is needed to show this), contradicting our hypothesis that $\{A,B\}$ is linearly dependent. Thus $\{A,B\}$ must be linearly independent.

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Kasra
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Kasra

Updated on November 20, 2022

Comments

  • Kasra
    Kasra 12 months

    can anybody help me plz? Let $A$ and $B$ be $n \times n$ matrices ($A$ and $B$ are not $0$) . Prove that if $A$ is symmetric and $B$ be skew-symmetric , then $\{A,B\}$ is a linearly independent set.

    • Erick Wong
      Erick Wong over 7 years
      For instance, have you tried writing down the definition of linearly independent for this particular set? If so, show us what you did. If not, then start by doing that, and if you're unable to then your question should be "what does linearly independent mean?", not "how do I do this problem?".
  • Kasra
    Kasra over 7 years
    I can't understand why the second equation is equal to zero. can you please clarify it? I know the concept of symmetric and skew-symmetric, I only can't figure out why it is zero. Thanks
  • Aweygan
    Aweygan over 7 years
    You take the transpose of everything in the first equation.
  • Cit5
    Cit5 over 2 years
    i dont see how you got aA=-bB. where was that derived from? So far, all the directions lead me to the second equation: aA=bB
  • Community
    Community over 1 year
    As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center.