Dimension theorem for vector spaces

Consider the subspace of {0,1}^n:

Is the below subsequence a basis of it?


(Keeping in mind that this space is defined over the field {0,1})

It seems to me, that it is a basis; but surely it can't be - because it has a dimension of 5, whereas
is also a valid basis, but has a dimension of 4.

(And by the Dimension theorem, all basis of any vector space must have equal cardinality).

What am I overlooking?

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This is real quick, but it seems that R4 can be formed by (1,0,1,0) can be formed by (1,0,0,1) - (0,1,0,1) + (0,1,1,0).  Which would mean it's not linearly independent from the other vectors, correct?
Ignore R4, it was a row reference I used in my calculations =P

The above should read:

"This is real quick, but it seems that (1,0,1,0) can be formed by..."

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InteractiveMindAuthor Commented:
But it's over the field {0,1}; you cannot use -1 ...
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Ah, right...I overlooked that =)
ee's original answer is correct, because in that field + is its own inverse, so
(1,0,0,1) + (0,1,0,1) + (0,1,1,0) = (1,0,1,0)
InteractiveMindAuthor Commented:
Thanks.  Sorry for the delay.
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