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Multiplying matrix units and standard basis vectors

  1. Dec 13, 2013 #1
    Hello all, I don't have a question on homework specifically, but I need clarification on something I'm reading in the textbook.

    I will be starting an abstract algebra class in the spring and it's been quite a few years since I've had linear algebra, so I'll be reviewing that material before the abstract algebra class starts. I've also started the first chapter of the book, Algebra, by Michael Artin. So far, I've been able to make sense of most of what I've read, but I'm stuck on this one formula at the top of page 10.

    He states that the formulas for multiplying matrix units and standard basis vectors are:

    eijej = ei

    and

    eijek = 0 if j ≠ k

    I understand from the previous page that the matrix unit eij has a 1 in the ij position as its only non-zero entry and based on an example from the previous page, you can show a standard m x n matrix as a linear combination that includes eij.

    My confusion is in figuring out what the ei and ej are in the above formula. The bottom of the previous page discusses a column vector ei but I wasn't sure how to connect this vector to the above formula or how I should multiply it with eij to get ej.

    Maybe if I saw a couple of concrete examples of what ei and ej are as compared with eij, it would help me clear up this confusion.

    Thanks for any help on this question!
     
  2. jcsd
  3. Dec 13, 2013 #2

    jbunniii

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    ##e_{i}## is a column vector with 1 in the ##i##th row, and zeros everywhere else.

    Simple example with a 4x4 matrix and 4x1 vectors:

    $$\left(\begin{array}{cccc}0&0&0&0\\
    0&0&1&0\\
    0&0&0&0\\
    0&0&0&0\end{array}\right)
    \left(\begin{array}{c}0\\0\\1\\0\end{array}\right) =
    \left(\begin{array}{c}0\\1\\0\\0\end{array}\right)$$
    which is the same as ##e_{23} e_3 = e_2##
     
  4. Dec 13, 2013 #3

    D H

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    They're vectors. In particular, they are column vectors. Right up front, Artin defines vectors as being either equivalent to a 1xn or an nx1 matrix. Think of ej as meaning ej1, with the column index of 1 implied by the fact that this N vector is equivalent to an Nx1 matrix.
     
  5. Dec 13, 2013 #4
    Thank you both! The matrix example and the additional column vector description was very helpful as I can now picture what's going on and how the matrix and vectors are indexed in reference to each other. Much clearer now!
     
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