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I don't understand how this is equal.

enter image description here

I can see three pivots so that makes it a rank 3 regarding the columns. Hence, the first three columns are pivot columns, so the matrix has a column rank 3.

But how is the row rank equal to three? To have a row rank of three, shouldn't the entire column 4 consist of [0,0,0,0]? Could someone explain to me what I am missing here?

For me, the definition of a pivot = [1,0,0,0] (so having a one and all others are zero), but when I'd check the row, I can see a [1,0,0,4], which is not equal to what I stated above!

Can someone help me comprehend it? Clearly I am missing something here...

Siyah
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    row rank = column rank. All three rows a linearly independent. – user29418 Mar 05 '17 at 18:16
  • What is your definition of "row rank"? – hmakholm left over Monica Mar 05 '17 at 18:16
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    If the three row vectors were linearly dependent, then so were their projections from $\mathbb R^3 \to \mathbb R^4$. – Santiago Mar 05 '17 at 18:16
  • I still don't get it. Could someone explain this me a bit simpler or with an example? My definition of a row rank is to check it as I check it in the column rank... I think that pivot = rank and clearly, that 4 on the end suggests it's not a pivot anymore, thus I can't see how the rank is being a 3... – Siyah Mar 05 '17 at 18:22
  • Related, and maybe helpful: http://math.stackexchange.com/questions/332908/looking-for-an-intuitive-explanation-why-the-row-rank-is-equal-to-the-column-ran – hlt Mar 05 '17 at 18:57

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