Rank and Nullity

Various Subspaces

Row Space, Column Space, Null Space

Let $A$ be an $m \times n$ matrix over $\newcommand{\R}{\mathbb{R}}\newcommand{\C}{\mathbb{C}}F\in \{\R,\C\}$ .
The row space of $A$ , denoted $row (A)$ , is the subspace of $F^{n}$ spanned by the row vectors of $A$ .
The column space of $A$ , denoted $col (A)$ , is the subspace of $F^{m}$ spanned by the column vectors of $A$ .
The null space of $A$ , denoted $null (A)$ or $ker (A)$ , is the subspace of $F^{n}$ consisting of all solutions to the homogeneous equation $A x = 0$ .

We immediately have the following characterization of solutions to linear equations:

Proposition

The equation $A x = b$ is consistent iff $b \in \col (A)$ .
If $x_{0}$ is a particular solution to $A x = b$ , then the complete solution set is $x_{0} + \null (A)$ .

Rank

Theorem

The row space and column space of a matrix have the same dimension (as subspaces of $F^{n}$ ).

Rank & Nullity

The rank of a matrix $A$ is the dimension of $\row (A)$ or $\col (A)$ .
The nullity of a matrix $A$ is the dimension of $\null (A)$ .

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;

$r r e f (A) = I$ ;

$A$ 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 $F^{n}$ ;

$r ank (A) = n$ ;

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

Quartz 5

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Rank and Nullity

Various Subspaces

Rank

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