Column space, null space and rank of a matrix

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

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

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

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

\(m\times n\) 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

Quiz