Rank and Nullity

Theorem

For an $n \times n$ matrix $A \in M_{n} (F)$ , the followings are equivalent:

$A$ is invertible;

$A x = 0$ has only the trivial solution;

$rre f (A) = I$ ;

$A$ is a product of elementary matrices;

$A x = b$ is consistent for every $b \in F^{n}$ ;

$A x = b$ has exactly one solution for every $b \in F$ ;

$det (A) \neq = 0$ ;

The column vectors of $A$ are linearly independent;

The row vectors of $A$ are linearly independent;

The column vectors of $A$ span $F^{n}$ ;

The row vectors of $A$ span $F^{n}$ ;

The column vectors of $A$ is a basis for $F^{n}$ ;

The row vectors of $A$ is a basis for $F^{n}$ ;

$r ank (A) = n$ ;

$n u ll i t y (A) = 0$ .

Quartz 4

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