Theorems on Matrix Representation of Linear Transformations

Addition and Scalar Multiplication of Linear Transformations #

Theorem-1: \(\bf [S+T]_{\alpha}^{\beta}=[S]_{\alpha}^{\beta}+[T]_{\alpha}^{\beta}\) #

  Statement:  Let \(\bf V \) and \(\bf W \) be two finite dimensional vector space over a same field \(\bf F \) wtih ordered bases \(\bf \alpha \) and \(\bf \beta \) respectively. If \(\bf S:V \to W\) and \(\bf T:V \to W\) are two linear transformation then \(\bf [S+T]_{\alpha}^{\beta}=[S]_{\alpha}^{\beta}+[T]_{\alpha}^{\beta}\) . \( \\ \)  Proof:  Given that \(\bf V \) and \(\bf W \) are two finite dimensional vector space over a same field \(\bf F \) wtih ordered bases \(\bf \alpha \) and \(\bf \beta \) respectively.\( \\ \)   Let \(\bf S:V \to W\) and \(\bf T:V \to W\) be two linear transformation.\( \\ \)   To Prove  \(\fcolorbox{red}{white}{\bf\(\bf [S+T]_{\alpha}^{\beta}=[S]_{\alpha}^{\beta}+[T]_{\alpha}^{\beta}\) }\) \( \\ \)   Let \(\bf dim~V=n\) and \(\bf dim~W=m\) \( \\ \)   and \(\bf \alpha=\{\alpha_{1},\alpha_{2},…\alpha_{n}\}\) and \(\beta=\{\beta_{1},\beta_{2},…\beta_{m}\}\) are ordered bases of \(\bf V \) and \(\bf W \) respectively. \( \\ \)   Then there exists unique scalars \(\bf a_{\textcolor{green}{~i}\textcolor{red}{~j}} \) and \(\bf b_{\textcolor{green}{~i}\textcolor{red}{~j}} \) in \(F \) for \(\bf \textcolor{green}{~i}=1,2,…,m \) and \(\bf \textcolor{red}{~j}=1,2,…,n \), such that\( \\ \)   \(\bf [S]_{\alpha}^{\beta} =[a_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\) and \(\bf [T]_{\alpha}^{\beta} =[b_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m}\times \textcolor{red}{~n}}\) \( \\ \)  i.e., \(\bf S(\alpha_{\textcolor{red}{~j}})=\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf a_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}} \) and \(\bf T(\alpha_{\textcolor{red}{~j}})=\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf b_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}} \) \( \\ \)

 

where $$\bf [a_{ij}]_{m\times n}= \begin{pmatrix} \bf a_{11} & \bf a_{12} & \bf … & \bf a_{1n} \\ \bf a_{21} & \bf a_{22} & \bf … & \bf a_{2n} \\ \bf …& \bf …& \bf … & \bf …\\ \bf a_{m1} & \bf a_{m2} &\bf … & \bf a_{mn} \\ \end{pmatrix}$$ and $$\bf [b_{ij}]_{m\times n} = \begin{pmatrix} \bf b_{11} & \bf b_{12} & \bf … & \bf b_{1n} \\ \bf b_{21} & \bf b_{22} & \bf … & \bf b_{2n} \\ \bf …& \bf …& \bf … & \bf …\\ \bf b_{m1} & \bf b_{m2} &\bf … & \bf b_{mn} \\ \end{pmatrix}$$

 Therefore \( \textcolor{red}{~j}=1,2,…,n \) \begin{align*} \bf (S+T)(\alpha_{\textcolor{red}{~j}}) & \bf = S(\alpha_{\textcolor{red}{~j}}) +T(\alpha_{\textcolor{red}{~j}}) \\ & \bf =\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf a_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}}+ \displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf b_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}}\\ & \bf =\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf (a_{\textcolor{green}{~i}\textcolor{red}{~j}}+b_{\textcolor{green}{~i}\textcolor{red}{~j}})~\beta_{\textcolor{green}{~i}}\\ \implies \bf [S+T]_{\alpha}^{\beta} & \bf = [a_{\textcolor{green}{~i}\textcolor{red}{~j}}+b_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\\ & \bf = [a_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}+[b_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\\ & \bf = [S]_{\alpha}^{\beta}+[T]_{\alpha}^{\beta} \end{align*}   Hence \(\fcolorbox{red}{white}{\bf\(\bf [S+T]_{\alpha}^{\beta}=[S]_{\alpha}^{\beta}+[T]_{\alpha}^{\beta}\) }\)

Theorem-2: \(\bf [cT]_{\alpha}^{\beta}=c[T]_{\alpha}^{\beta}\) #

  Statement:  Let \(\bf V \) and \(\bf W \) be two finite dimensional vector space over a same field \(\bf F \) wtih ordered bases \(\bf \alpha \) and \(\bf \beta \) respectively. If \(\bf T:V \to W\) is an linear transformation then \(\bf [cT]_{\alpha}^{\beta}=c[T]_{\alpha}^{\beta}\) for any \( \bf c\in F\). \( \\ \)  Proof:  Given that \(\bf V \) and \(\bf W \) are two finite dimensional vector space over a same field \(\bf F \) wtih ordered bases \(\bf \alpha \) and \(\bf \beta \) respectively.\( \\ \)   Let \(\bf T:V \to W\) be an linear transformation.\( \\ \)   To Prove  \(\fcolorbox{red}{white}{\bf\(\bf [cT]_{\alpha}^{\beta}=c[T]_{\alpha}^{\beta}~~ \forall~c\in F\)}\) \( \\ \)   Let \( \bf c\in F\) \( \\ \)   Let \(\bf dim~V=n\) and \(\bf dim~W=m\) \( \\ \)   Let \(\bf \alpha=\{\alpha_{1},\alpha_{2},…\alpha_{n}\}\) and \(\beta=\{\beta_{1},\beta_{2},…\beta_{m}\}\) are ordered bases of \(\bf V \) and \(\bf W \) respectively. \( \\ \)   Then there exists unique scalars \(\bf b_{\textcolor{green}{~i}\textcolor{red}{~j}} \) in \(F \) for \(\bf \textcolor{green}{~i}=1,2,…,m \) and \(\bf \textcolor{red}{~j}=1,2,…,n \), such that\( \\ \)  \(\bf [T]_{\alpha}^{\beta} =[b_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m}\times \textcolor{red}{~n}}\) \( \\ \)  i.e., \(\bf T(\alpha_{\textcolor{red}{~j}})=\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf b_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}} \) \( \\ \)  Therefore \( \textcolor{red}{~j}=1,2,…,n \) and \( \bf c\in F\), \begin{align*} \bf (cT)(\alpha_{\textcolor{red}{~j}}) & \bf = c\left[ T(\alpha_{\textcolor{red}{~j}})\right] \\ & \bf =c\left[ \displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf b_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}} \right]\\ & \bf =\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf (cb_{\textcolor{green}{~i}\textcolor{red}{~j}})~\beta_{\textcolor{green}{~i}}\\ \implies \bf [S+T]_{\alpha}^{\beta} & \bf = [cb_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\\ & \bf = c[b_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\\ & \bf = c[T]_{\alpha}^{\beta} \end{align*}  Since \(\bf c\) be arbitrary scalar of \( \bf F\), then \(\fcolorbox{red}{white}{\bf\(\bf [cT]_{\alpha}^{\beta}=c[T]_{\alpha}^{\beta}~~\forall~c\in F \)}\).

Theorem-3: \(\bf [cS+dT]_{\alpha}^{\beta}=c[S]_{\alpha}^{\beta}+d[T]_{\alpha}^{\beta}\) #

  Statement:  Let \(\bf V \) and \(\bf W \) be two finite dimensional vector space over a same field \(\bf F \) wtih ordered bases \(\bf \alpha \) and \(\bf \beta \) respectively. If \(\bf S:V \to W\) and \(\bf T:V \to W\) are two linear transformation then \(\bf [cS+dT]_{\alpha}^{\beta}=[cS]_{\alpha}^{\beta}+[dT]_{\alpha}^{\beta}\) for any \( \bf c,d\in F\). \( \\ \)  Proof:  Given that \(\bf V \) and \(\bf W \) are two finite dimensional vector space over a same field \(\bf F \) wtih ordered bases \(\bf \alpha \) and \(\bf \beta \) respectively.\( \\ \)   Let \(\bf S:V \to W\) and \(\bf T:V \to W\) be two linear transformation.\( \\ \)   To Prove  \(\fcolorbox{red}{white}{\bf \(\bf [cS+dT]_{\alpha}^{\beta}=[cS]_{\alpha}^{\beta}+[dT]_{\alpha}^{\beta}~~ \forall~c,d\in F\)}\) \( \\ \)   Let \( \bf c,d\in F\) \( \\ \)   And \(\bf dim~V=n\) and \(\bf dim~W=m\). \( \\ \)   Also \(\bf \alpha=\{\alpha_{1},\alpha_{2},…\alpha_{n}\}\) and \(\beta=\{\beta_{1},\beta_{2},…\beta_{m}\}\) are ordered bases of \(\bf V \) and \(\bf W \) respectively. \( \\ \)   Then there exists unique scalars \(\bf a_{\textcolor{green}{~i}\textcolor{red}{~j}} \) and \(\bf b_{\textcolor{green}{~i}\textcolor{red}{~j}} \) in \(F \) for \(\bf \textcolor{green}{~i}=1,2,…,m \) and \(\bf \textcolor{red}{~j}=1,2,…,n \), such that\( \\ \)   \(\bf [S]_{\alpha}^{\beta} =[a_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\) and \(\bf [T]_{\alpha}^{\beta} =[b_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m}\times \textcolor{red}{~n}}\) \( \\ \)  i.e., \(\bf S(\alpha_{\textcolor{red}{~j}})=\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf a_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}} \) and \(\bf T(\alpha_{\textcolor{red}{~j}})=\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf b_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}} \) \( \\ \)

 Therefore \( \textcolor{red}{~j}=1,2,…,n \) and \( \bf c,d\in F\), \begin{align*} \bf (cS+dT)(\alpha_{\textcolor{red}{~j}}) & \bf = (cS)(\alpha_{\textcolor{red}{~j}}) +(dT)(\alpha_{\textcolor{red}{~j}}) \\ & \bf = c\left[S(\alpha_{\textcolor{red}{~j}})\right] +d\left[T(\alpha_{\textcolor{red}{~j}})\right] \\ & \bf =c\left[ \displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf a_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}} \right]+ d\left[ \displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf b_{\textcolor{green}{~i}\textcolor{red}{~j}}~\beta_{\textcolor{green}{~i}}\right]\\ & \bf = \displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf (ca_{\textcolor{green}{~i}\textcolor{red}{~j}})~\beta_{\textcolor{green}{~i}} + \displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf (db_{\textcolor{green}{~i}\textcolor{red}{~j}})~\beta_{\textcolor{green}{~i}}\\ & \bf =\displaystyle\sum_{\textcolor{green}{~i}=1}^{m} \bf (ca_{\textcolor{green}{~i}\textcolor{red}{~j}}+db_{\textcolor{green}{~i}\textcolor{red}{~j}})~\beta_{\textcolor{green}{~i}}\\ \implies \bf [cS+dT]_{\alpha}^{\beta} & \bf = [ca_{\textcolor{green}{~i}\textcolor{red}{~j}}+db_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\\ & \bf = [ca_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}+[db_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\\ & \bf = c[a_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}+d[b_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\\ & \bf = c[S]_{\alpha}^{\beta}+d[T]_{\alpha}^{\beta} \end{align*}   Since \(\bf c,d\) be arbitrary scalar of \( \bf F\), then \(\fcolorbox{red}{white}{\bf\(\bf [cS+dT]_{\alpha}^{\beta}=c[S]_{\alpha}^{\beta}+d[T]_{\alpha}^{\beta}~~\forall~c,d\in F\)}\).

Composition of Linear Transformation #

Theorem-4: \(\bf [TS]_{\alpha}^{\gamma}=[T]_{\beta}^{\gamma}[S]_{\alpha}^{\beta}\) #

  Statement:  Let \(\bf V \), \(\bf W \) and \(\bf U \) be three finite dimensional vector space over a same field \(\bf F \) wtih ordered bases \(\bf \alpha \), \(\bf \beta \) and \(\bf \gamma \) respectively. If \(\bf S:V \to W\) and \(\bf T:W\to U\) are two linear transformation then \(\bf [TS]_{\alpha}^{\gamma}=[T]_{\beta}^{\gamma}[S]_{\alpha}^{\beta}\) . \( \\ \)  Proof:  Given that \(\bf V \), \(\bf W \) and \(\bf U \) are three finite dimensional vector space over a same field \(\bf F \) wtih ordered bases \(\bf \alpha \), \(\bf \beta \) and \(\bf \gamma \) respectively.\( \\ \)   Let \(\bf S:V \to W\) and \(\bf T:W\to U\) be two linear transformation. Then \(\bf TS:V \to U\) is also an linear transformation and \(\bf [TS]_{\alpha}^{\gamma} \) exists.\( \\ \)  To Prove  \(\fcolorbox{red}{white}{\bf\(\bf [TS]_{\alpha}^{\gamma}=[T]_{\beta}^{\gamma}[S]_{\alpha}^{\beta}\) }\) \( \\ \)   Let \(\bf dim~V=n\), \(\bf dim~W=m\) and \(\bf dim~U=p\) \( \\ \)   and \(\bf \alpha=\{\alpha_{1},\alpha_{2},…\alpha_{n}\}\), \(\bf \beta=\{\beta_{1},\beta_{2},…\beta_{m}\}\) and \(\bf \gamma=\{\gamma_{1},\gamma_{2},…\gamma_{p}\}\) are ordered bases of \(\bf V \), \(\bf W \) and \(\bf U \) respectively. \( \\ \)   Then there exists unique scalars \(\bf a_{\textcolor{green}{~k}\textcolor{red}{~l}} \) and \(\bf b_{\textcolor{green}{~i}\textcolor{red}{~j}} \) in \(F \) for \(\bf \textcolor{green}{~k}=1,2,…,m \), \(\bf \textcolor{red}{~l}=1,2,…,n \), \(\bf \textcolor{red}{~i}=1,2,…,p \) and \(\bf \textcolor{red}{~j}=1,2,…,m \), such that\( \\ \)   \(\bf [S]_{\alpha}^{\beta} =[a_{\textcolor{green}{~k}\textcolor{red}{~l}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\) and \(\bf [T]_{\beta}^{\gamma} =[b_{\textcolor{green}{~i}\textcolor{red}{~j}}]_{\textcolor{green}{~p}\times \textcolor{red}{~m}}\) \( \\ \)  i.e., \(\bf S(\alpha_{\textcolor{red}{~l}})=\displaystyle\sum_{\textcolor{green}{~k}=1}^{m} \bf a_{\textcolor{green}{~k}\textcolor{red}{~l}}~\beta_{\textcolor{green}{~k}} \) and \(\bf T(\beta_{\textcolor{red}{~j}})=\displaystyle\sum_{\textcolor{green}{~i}=1}^{p} \bf b_{\textcolor{green}{~i}\textcolor{red}{~j}}~\gamma_{\textcolor{green}{~i}} \) \( \\ \)

 Therefore \( \textcolor{red}{~l}=1,2,…,n \) \begin{align*} \bf (TS)(\alpha_{\textcolor{red}{~l}}) & \bf = T(S(\alpha_{\textcolor{red}{~l}})) \\ & \bf = T\left(\displaystyle\sum_{\textcolor{green}{~k}=1}^{m} \bf a_{\textcolor{green}{~k}\textcolor{red}{~l}}~\beta_{\textcolor{green}{~k}}\right) \\ & \bf = \displaystyle\sum_{\textcolor{green}{~k}=1}^{m} \bf a_{\textcolor{green}{~k}\textcolor{red}{~l}}~T\left(\beta_{\textcolor{green}{~k}}\right)~since~T~is~an~LT \\ & \bf = \displaystyle\sum_{\textcolor{green}{~k}=1}^{m} \bf a_{\textcolor{green}{~k}\textcolor{red}{~l}}~\displaystyle\sum_{\textcolor{green}{~i}=1}^{p} \bf b_{\textcolor{green}{~i}\textcolor{red}{~k}}~\gamma_{\textcolor{green}{~i}}\\ & \bf = \displaystyle\sum_{\textcolor{green}{~i}=1}^{p} \displaystyle\sum_{\textcolor{green}{~k}=1}^{m} \bf b_{\textcolor{green}{~i}\textcolor{red}{~k}}a_{\textcolor{green}{~k}\textcolor{red}{~l}}~~\gamma_{\textcolor{green}{~i}}\\ \implies \bf [TS]_{\alpha}^{\gamma} & \bf = \left[\displaystyle\sum_{\textcolor{green}{~k}=1}^{m} \bf b_{\textcolor{green}{~i}\textcolor{red}{~k}}a_{\textcolor{green}{~k}\textcolor{red}{~l}}\right]_{\textcolor{green}{~p} \times \textcolor{red}{~n}}\\ & \bf =[b_{\textcolor{green}{~i}\textcolor{red}{~k}}]_{\textcolor{green}{~p}\times \textcolor{red}{~m}}[a_{\textcolor{green}{~k}\textcolor{red}{~l}}]_{\textcolor{green}{~m} \times \textcolor{red}{~n}}\\ & \bf = [T]_{\beta}^{\gamma}[S]_{\alpha}^{\beta} \end{align*}   Hence \(\fcolorbox{red}{white}{\bf \(\bf [TS]_{\alpha}^{\gamma}=[T]_{\beta}^{\gamma}[S]_{\alpha}^{\beta}\) }\)