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Thread: Derivation of determinants?

  1. #1
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    Derivation of determinants?

    Hi, I was wondering if anyone can point me in the right direction to understanding determinants. I don't want to know how to evaluate them. I want to know why they are evaluated the way they are. I want to know how they were derived and what the answers mean. Any links or replies regarding this topic would be great. I have been unable to find any information on this strangely.

    Thank you,

    -Andrew
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ajohnson388 View Post
    Hi, I was wondering if anyone can point me in the right direction to understanding determinants. I don't want to know how to evaluate them. I want to know why they are evaluated the way they are. I want to know how they were derived and what the answers mean. Any links or replies regarding this topic would be great. I have been unable to find any information on this strangely.

    Thank you,

    -Andrew
    Where the determinant came from? Behind the scenes the determinant came from noticing that if $\displaystyle T$ is an endomorphism on some $\displaystyle n$-dimensional $\displaystyle F$-space $\displaystyle V$ and $\displaystyle K:V\boxplus\cdots\boxplus V\to F$ is some alternating $\displaystyle n$-linear form then the function $\displaystyle (T\odot K)(v_1,\cdots,v_n)=K(T(v_1),\cdots,T(v_n))$ is an alternating $\displaystyle n$-linear form. But, we know that the space of alternating $\displaystyle n$-linear forms has dimension $\displaystyle \displaystyle {n\choose n}=1$. In particular $\displaystyle T\odot K=\delta_T K$ for some constant $\displaystyle \delta_T$. We then define $\displaystyle \delta_T$ to be the determinant of $\displaystyle T$. It's useful because it gives quantatative measure of invertibility. Namely, $\displaystyle T\in\text{GL}\left(V\right)$ if and only if $\displaystyle \delta_T\ne 0$ (or more generally if dealing with commutative rings if $\displaystyle \delta_T$ is a unit). Does that at least give you start?
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  3. #3
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    You might take a look at Chapter 4 on this web site:

    Linear Algebra: Table of Contents
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    I think this is a nice introduction: http://www.math.brown.edu/~treil/papers/LADW/LADW.pdf
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    MHF Contributor Drexel28's Avatar
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    I'm sorry if I was a little too technical before. For a good purely matrix analysis approach to determinants none tops the first chapter or so of Shilov.
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    Thank you for all the information. I realize now I need to develop a strong knowledge in concrete mathematics before continuing to try to understand. I'm weak on the mathematical notations so it makes it hard to understand. At least now I have access to information on determinants, so I thank you all for that.

    @Drexel28
    I do own Shilov's book on Linear Algebra (well-written indeed), but I had trouble finding if he included the derivation of determinants or not. Whether or not it is revealed later in the book or I misread, I should still probably read on concrete mathematics before continuing.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ajohnson388 View Post
    Thank you for all the information. I realize now I need to develop a strong knowledge in concrete mathematics before continuing to try to understand. I'm weak on the mathematical notations so it makes it hard to understand. At least now I have access to information on determinants, so I thank you all for that.

    @Drexel28
    I do own Shilov's book on Linear Algebra (well-written indeed), but I had trouble finding if he included the derivation of determinants or not. Whether or not it is revealed later in the book or I misread, I should still probably read on concrete mathematics before continuing.
    Hmm, I guess you should clarify precisely what you mean by "derivation"?
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  8. #8
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    If by derivation you mean motivation, then no, Shilov gives none. I think on something like page 3 or 4 he just throws at you the general determinant formula in all its glory.

    The book I linked you to, Linear Algebra Done Wrong, motivates this definition by considering the "volume" formed by n vectors, each with n components (I quickly skimmed awkward's link and it looks similar). From this, Shilov's mysterious definition emerges and I agree with Drexel; I haven't seen a better treatment of determinants than Shilov's.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Drexel28 View Post
    Where the determinant came from? Behind the scenes the determinant came from noticing that if $\displaystyle T$ is an endomorphism on some $\displaystyle n$-dimensional $\displaystyle F$-space $\displaystyle V$ and $\displaystyle K:V\boxplus\cdots\boxplus V\to F$ is some alternating $\displaystyle n$-linear form then the function $\displaystyle (T\odot K)(v_1,\cdots,v_n)=K(T(v_1),\cdots,T(v_n))$ is an alternating $\displaystyle n$-linear form. But, we know that the space of alternating $\displaystyle n$-linear forms has dimension $\displaystyle \displaystyle {n\choose n}=1$. In particular $\displaystyle T\odot K=\delta_T K$ for some constant $\displaystyle \delta_T$. We then define $\displaystyle \delta_T$ to be the determinant of $\displaystyle T$. It's useful because it gives quantatative measure of invertibility. Namely, $\displaystyle T\in\text{GL}\left(V\right)$ if and only if $\displaystyle \delta_T\ne 0$ (or more generally if dealing with commutative rings if $\displaystyle \delta_T$ is a unit). Does that at least give you start?
    Honestly, I really don't think that's where the determinant came from.
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  10. #10
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    Honestly, I really don't think that's where the determinant came from.
    Of course this isn't where the determinant of a matrix originally came from. The definition of the determinant of a matrix dates all the way back to the 1850s where matrix theory's progenitors such as James Sylvester et. al. developed the notion of a determinant to find necessary and sufficient conditions for the solubility of a system of linear equations. That said, I rather loosely interpreted the question as where "should" the determinant have come from?
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