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QUICK QUESTION about matrix (linear algebra)

  1. Dec 17, 2011 #1
    Hello guys,
    I am wondering, could a elementary row operation be done to a matrix without changing it.

    I know that if we have the SYSTEM, then we could exchange rows, scalar multiplication of a row, linear combination of one rows.

    but if we are only given a matrix, and we are asked to row reduce that matrix, could we still do scalar multiplication of a rows, exchange rows, etc?

    Thank you
     
  2. jcsd
  3. Dec 17, 2011 #2

    vela

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    Yup. You can think of row-reducing a matrix by itself as essentially solving the system Ax=0, which is something you might want to do to show that vectors are linearly independent, that the kernel of a linear map is trivial, etc. There's really no point in tacking on a column of zeros to form an enhanced matrix since the row operations will leave it unchanged.
     
  4. Dec 17, 2011 #3
    Thank you for your reply.
    I remember my TA told me that when we are doing row reduction for a matrix, we must use this symbol "→", not "=", or else marks will be reduced.
    I ask him why, but I not exactly remember what he said.

    so, I got confuse, .... because of this symbol, sometimes when I am asked to reduce a matrix, I afraid of exchanging rows, etc, because I thought it's not a system.

    thank you, very helpful
     
  5. Dec 17, 2011 #4

    vela

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    Your TA said that because the equal sign has a specific meaning. If you write, for example,
    [tex]\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix},[/tex] it means that a=1, b=2, c=3, and d=4. If you have the matrix [itex]\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}[/itex] and you subtract row 1 from row 2, it wouldn't make sense to write
    [tex] \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}[/tex] because you'd be saying that 1=0 and 3=1, which is obviously nonsense.
     
  6. Dec 18, 2011 #5

    Mark44

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    The symbol for equivalence is used more often, I believe. In fact, I don't recall seeing an arrow used in this context.

    Using vela's example, we would have
    [tex]\begin{bmatrix}1&2 \\ 1 & 3\end{bmatrix} \equiv \begin{bmatrix}1&2 \\ 0 & 1\end{bmatrix}[/tex]

    Here the two matrices are equivalent.
    The first matrix could represent the system of equations
    x + 2y = 0
    x + 3y = 0

    The system above is equivalent to this system:
    x + 2y = 0
    0x + y = 0

    Notice that it wouldn't make any sense to call the two systems of equations "equal."
     
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