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Determine whether this matrix' columns are linearly dependent or not.

$$\begin{bmatrix} 1 & 0 & 2 \\ 0 & -1 & -2 \\ 2 & -2 & 0 \end{bmatrix}$$

The determinant is $0$ - therefore they are linearly dependent!

Without making any calculations.

Whoa there. How do you determine column dependency without calculating the determinant?

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    $\begingroup$ Add the first two columns. $\endgroup$ – user61527 Apr 9 '14 at 5:03
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Hint:

Write a linear combination of column 1 and 2, that is $2C_1 + 2 C_2$.

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  • $\begingroup$ You mean a combination of the form $aC_1 + bC_2 = C_3$? $\endgroup$ – Zol Tun Kul Apr 9 '14 at 5:06
  • $\begingroup$ Yeah, it makes sense. Although the question says "without any additional calculations" I suppose this is the right way (I don't think there's a way to determine this without doing any calculations). $\endgroup$ – Zol Tun Kul Apr 9 '14 at 5:09
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Note that the dimension of the column space is equal to that of the row space. Therefore, simply perform Gaussian elimination to determine the number of linearly independent rows (and hence the number of linearly independent columns).

In other words, the number of linearly independent columns will equal the number of pivots when your matrix has been row-reduced.

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