# Column space, null space and rank of a matrix

Aug-Nov 2020

## Recap

• Vector space $V$ over a scalar field $F$
• $F$: real field $\mathbb{R}$ or complex field $\mathbb{C}$ in this course
• Linear map $T:V\to W$
• $T(au+bv)=aT(u)+bT(v)$
• Matrix of linear map with respect to bases for $V$ and $W$
• Basis for $V$: $\{v_1,\ldots,v_n\}$
• Column $j$: coordinates of $T(v_j)$ with respect to basis of $W$
• null $T=\{v\in V:Tv=0\}$, range $T=\{Tv:v\in V\}$
• Fundamental theorem: dim null $T$ + dim range $T$ = dim $V$

## From $T:F^n\to F^m$ to an $m\times n$ matrix

Basis for $F^n$: $\{v_1,\ldots,v_n\}$, basis for $F^m$: $\{w_1,\ldots,w_m\}$

Matrix for $T$: $A=\begin{bmatrix} A_{11}&A_{12}&\cdots&A_{1n}\\ A_{21}&A_{22}&\cdots&A_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ A_{m1}&A_{m2}&\cdots&A_{mn} \end{bmatrix}$

For input $v=a_1v_1+\cdots+a_nv_n\leftrightarrow\begin{bmatrix}a_1\\\vdots\\a_n\end{bmatrix}$,

\begin{align} Tv&=a_1\quad Tv_1\ +\cdots+a_n\quad Tv_n\\[5pt] &=a_1\begin{bmatrix}A_{11}\\A_{21}\\\vdots\\A_{m1}\end{bmatrix}+\cdots+a_n\begin{bmatrix}A_{1n}\\A_{2n}\\\vdots\\A_{mn}\end{bmatrix} \quad\leftrightarrow A\begin{bmatrix}a_1\\a_2\\\vdots\\a_n\end{bmatrix} \end{align}

Leads to definition of matrix-vector product

## From an $m\times n$ matrix to $T:F^n\to F^m$

$A=\begin{bmatrix} A_{11}&A_{12}&\cdots&A_{1n}\\ A_{21}&A_{22}&\cdots&A_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ A_{m1}&A_{m2}&\cdots&A_{mn} \end{bmatrix}$

• Given an $m\times n$ matrix $A$, fix bases for $F^n$ and $F^m$

• $\{v_1,\ldots,v_n\}$ and $\{w_1,\ldots,w_m\}$

• say, standard bases

• $j$-th column of $A$: $Tv_j$ expressed in $\{w_1,\ldots,w_m\}$

• Input vector $v$: expressed in $\{v_1,\ldots,v_n\}$ as a column vector

• Output $Tv$: matrix-vector product

## Column space and null space of a matrix

$m\times n$ matrix $A=\begin{bmatrix} A_{11}&A_{12}&\cdots&A_{1n}\\ A_{21}&A_{22}&\cdots&A_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ A_{m1}&A_{m2}&\cdots&A_{mn} \end{bmatrix}$ $\leftrightarrow$ $T:F^n\to F^m$

• null $T$
• $\{x\in F^n: Ax=0\}$
• solutions to homogeneous equations $Ax=0$
• called null space of matrix $A$
• nullity: dimension of null space
• range $T$
• $\{Ax: x\in F^n\}$
• span of columns of $A$
• called column space of $A$, denoted colspace $A$
• column rank or rank: dimension of column space

By fundamental theorem of linear maps, $n=\text{rank}(A)+\text{nullity}(A)$.

## Examples: $2\times 2$, represents $T:F^2\to F^2$

$A=\begin{bmatrix}0&0\\0&0\end{bmatrix}$

• colspace $A = \{0\}$, rank = 0
• null $A = F^2$, nullity = 2

$A=\begin{bmatrix}1&0\\2&0\end{bmatrix}$

• colspace $A = \text{span}\{(1,2)\}$, rank = 1
• null $A = \text{span}\{(0,1)\}$, nullity = 1

$A=\begin{bmatrix}1&3\\2&6\end{bmatrix}$

• colspace $A = \text{span}\{(1,2)\}$, rank = 1
• null $A = \text{span}\{(-3,1)\}$, nullity = 1

Rank 1

• At least one non-zero column
• One column is a multiple of other column

## Examples: $2\times 2$, represents $T:F^2\to F^2$

$A=\begin{bmatrix}1&0\\0&1\end{bmatrix}$

• colspace $A = F^2$, rank = 2
• null $A = \{0\}$, nullity = 0

$A=\begin{bmatrix}1&3\\2&4\end{bmatrix}$

• colspace $A = \text{span}\{(1,2),(3,4)\}$, rank = 2
• null $A = \{0\}$, nullity = 0

Rank 2

• colspace = $F^2$, null space = $\{0\}$
• columns are linearly independent
• called full rank or full column rank

## Examples: $3\times 2$, represents $T:F^2\to F^3$

$A=\begin{bmatrix}1&3\\2&6\\3&9\end{bmatrix}$

• colspace $A = \text{span}\{(1,2,3)\}$, rank = 1
• null $A=\text{span}\{(-3,1)\}$, nullity = 1

Rank 1

• At least one non-zero column
• One column is a multiple of other column

$A=\begin{bmatrix}1&3\\2&4\\3&5\end{bmatrix}$

• colspace $A = \text{span}\{(1,2,3),(3,4,5)\}$, rank = 2
• null $A=\{0\}$, nullity = 0

Rank 2

• null space = $\{0\}$
• col space = dim 2 subspace of $F^3$

## Examples: $2\times 3$, represents $T:F^3\to F^2$

$A=\begin{bmatrix}1&3&5\\2&6&10\end{bmatrix}$

• colspace $A = \text{span}\{(1,2)\}$, rank = 1
• null $A=\text{span}\{(-3,1,0),(-5,0,1)\}$, nullity = 2

Rank 1

• At least one non-zero column
• Columns are multiples of non-zero column

$A=\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}$

• colspace $A = F^2$, rank = 2
• null $A=\text{span}\{(1,-2,1)\}$, nullity = 1

Rank 2

• col space = $F^2$
• null space, rank 1

## $m\times n$ matrix representing $T:F^n\to F^m$

col space = span{columns}

• use Gaussian elimination to reduce and find dimension

null space

• use fundamental theorem to find dimension
• solve homogeneous linear equation to find basis
• dim colspace $\le m$, dim null $\le n$

• dim colspace $\le n$

• $m<n$

• dim null $> 0$
• Not injective
• $m>n$

• dim colspace $< m$
• Not surjective