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If the $N(A)$ has more than just the $\vec{0}$, does that mean that the $C(A)$ is linearly dependant?

Moreover, I just started with Linear Algebra a week ago, so if there is something wrong about my notation of the null space, column vectors etc. Tell me please!

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I am guessing that $N(A)$ is the null space, what is $C(A)$, the columns? –  copper.hat Oct 10 '12 at 19:57
Column vectors of Matrix A. –  ZafarS Oct 10 '12 at 19:58
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1 Answer

up vote 2 down vote accepted

Assuming that $N,C$ refer to the null space and columns respectively, then yes.

If $Ax = 0$, with $x \neq 0$, then this is equivalent to $\sum x_i a_i = 0$, with at least one $x_i \neq 0$, where $a_i$ is the $i$th column of $A$.

Hence the columns of $A$ are linearly dependent iff the null space of $A$ contains a non-zero vector.

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Thank you. If I may ask, and this will probably be a stupid question, why does the zero have a sort of comma in it? Is that the notation for the zero-vector? –  ZafarS Oct 10 '12 at 20:02
@ZafarS no it happens to me too sometimes, it is not the notation,it is an illusion, you have to adjust the zoom of the screen to see properly what it was written. –  clark Oct 10 '12 at 20:06
Ok, and a final small question (I don't think it is worth asking seperately) is this statement correct: Nullity (A) = # of free variables in rref(A)? –  ZafarS Oct 10 '12 at 20:07
I presume rref is reduced row echelon form. The answer is yes. –  copper.hat Oct 10 '12 at 20:21
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