Algebraic operations on linear maps

Andrew Thangaraj

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
  • 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 \(V\)
  • \(m\times n\) matrix A represents a linear map \(T:F^n\to F^m\)
    • colspace(A) = range T, null(A) = null(T)

Addition and scalar multiplication of linear maps

\(S,T:V\to W\) are linear maps, \(\lambda\in F\). Sum of \(S\) and \(T\), denoted \(S+T\), is defined as \[(S+T)v = Sv + Tv.\] The scalar product of \(\lambda\) and \(T\), denoted \(\lambda T\), is defined as \[(\lambda T)v = \lambda (Tv).\]

Results

  • \(S+T\) and \(\lambda T\) are linear maps from \(V\to W\).

  • \(\mathcal{L}(V,W)\): set of all linear maps from \(V\) to \(W\)

    • \(\mathcal{L}(V,W)\) is a vector space over \(F\)

    • under addition and scalar multiplication defined above

    • additive identity: zero map

Product of linear maps

\(U,V,W\): vector spaces over \(F\) and \(T:U\to V\), \(S:V\to W\) are linear maps. Product of \(S\) and \(T\), denoted \(ST\), is defined as \[(ST)u = S(Tu).\] \(ST\): linear map from \(U\) to \(W\); composition of the two maps \(T\) and \(S\)

Algebraic properties of linear maps

  • Multiplication is associative

    • \((T_1T_2)T_3=T_1(T_2T_3)\)

  • Multiplication need not be commutative

    • If \(ST\) is defined, \(TS\) may not even be defined

    • Even if \(ST\) and \(TS\) are defined, they may not be the same map

  • Distributive property

    • \((S_1+S_2)T = S_1T + S_2T\) and \(S(T_1+T_2) = ST_1 + ST_2\)

Addition and scalar multiplication of matrices

\[\begin{align} \begin{bmatrix} A_{11}&\cdots&A_{1n}\\ \vdots&\ddots&\vdots\\ A_{m1}&\cdots&A_{mn} \end{bmatrix}+\begin{bmatrix} B_{11}&\cdots&B_{1n}\\ \vdots&\ddots&\vdots\\ B_{m1}&\cdots&B_{mn} \end{bmatrix}&=\begin{bmatrix} A_{11}{+}B_{11}&\cdots&A_{1n}{+}B_{1n}\\ \vdots&\ddots&\vdots\\ A_{m1}{+}B_{m1}&\cdots&A_{mn}{+}B_{mn} \end{bmatrix}\\[5pt] \lambda\begin{bmatrix} A_{11}&\cdots&A_{1n}\\ \vdots&\ddots&\vdots\\ A_{m1}&\cdots&A_{mn} \end{bmatrix}&=\begin{bmatrix} \lambda A_{11}&\cdots&\lambda A_{1n}\\ \vdots&\ddots&\vdots\\ \lambda A_{m1}&\cdots&\lambda A_{mn} \end{bmatrix} \end{align}\]

\(V,W\): finite-dimensional and \(S,T:V\to W\). Fix bases for \(V\) and for \(W\).

\(M(S), M(T), M(S+T), M(\lambda T)\) are matrices for \(S\), \(T\), \(S+T\) and \(\lambda T\) with respect to the chosen bases. Then, \[\begin{align} M(S+T)&=M(S)+M(T)\\ M(\lambda T)&=\lambda M(T) \end{align}\]

Matrix multiplication

\[\begin{bmatrix} A_{11}&\cdots&A_{1n}\\ \vdots&\ddots&\vdots\\ A_{m1}&\cdots&A_{mn} \end{bmatrix}\begin{bmatrix} B_{11}&\cdots&B_{1k}\\ \vdots&\ddots&\vdots\\ B_{n1}&\cdots&B_{nk} \end{bmatrix}=\begin{bmatrix} C_{11}&\cdots&C_{1k}\\ \vdots&\ddots&\vdots\\ C_{m1}&\cdots&C_{mk} \end{bmatrix}\]

\(C_{ij}=\sum_{l=1}^n A_{il}B_{lj}\)

(dot product of \(i\)-th row of \(A\) and \(j\)-th column of \(B\))

\(U,V,W\): finite-dimensional and \(T:U\to V\), \(S:V\to W\). Fix bases for \(U\), \(V\) and \(W\).

\(M(S), M(T), M(ST)\) are matrices for \(S\), \(T\) and \(ST\) with respect to the chosen bases. Then, \[M(ST)=M(S)M(T)\]

Interesting matrix multiplications

\(\begin{bmatrix} b_1&b_2&\cdots&b_n \end{bmatrix}\begin{bmatrix} a_1\\a_2\\\vdots\\a_n \end{bmatrix}=a_1b_1+\cdots+a_nb_n\)

  • \(S: F^n\to F\), \(T:F\to F^n\)

  • \(ST: F\to F\)

\(\begin{bmatrix} a_1\\a_2\\\vdots\\a_m \end{bmatrix}\begin{bmatrix} b_1&b_2&\cdots&b_n \end{bmatrix}=\begin{bmatrix} a_1b_1&a_1b_2&\cdots&a_1b_n\\ a_2b_1&a_2b_2&\cdots&a_2b_n\\ \vdots&\vdots&\vdots&\vdots\\ a_mb_1&a_mb_2&\cdots&a_mb_n \end{bmatrix}\)

  • \(S:F\to F^m\), \(T:F^n\to F\)

  • \(ST: F^n\to F^m\)

More on matrix multiplication

\[\begin{bmatrix} A_{11}&\cdots&A_{1n}\\ \vdots&\ddots&\vdots\\ A_{m1}&\cdots&A_{mn} \end{bmatrix}\begin{bmatrix} B_{11}&\cdots&B_{1k}\\ \vdots&\ddots&\vdots\\ B_{n1}&\cdots&B_{nk} \end{bmatrix}=\begin{bmatrix} C_{11}&\cdots&C_{1k}\\ \vdots&\ddots&\vdots\\ C_{m1}&\cdots&C_{mk} \end{bmatrix}\]

\(C_{ij}=\sum_{l=1}^n A_{il}B_{lj}\)

  • \(C_{ij}\): (\(i\)-th row of \(A\)) \(\times\) (\(j\)-th column of \(B\))

  • \(i\)-th row of \(C\): (\(i\)-th row of \(A\)) \(\times\) \(B\)

  • \(j\)-th column of \(C\): \(A\) \(\times\) (\(j\)-th column of \(B\))

  • \(C=\sum_{l=1}^n (l\)-th column of \(A) \times (l\)-th row of \(B)\)