Linear Tansformations
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Matrix Representation of Linear Transformations
Last Updated: June 20, 2024Ordered Basis Definition: Let \(\bf V \) be finite dimensional vector space over a field \(\bf F \). An \(\fcolorbox{red}{white}{\bf ordered basis}\) for \(\bf V\) is a basis of \(\bf V \) provided with a specific order. Example Let us consider the vector space \(\bf \real^{3} \) and let \(\bf \alpha_{1}=(1,0,0) \), \(\bf \alpha_{2}=(0,1,0)...
Theorems on Matrix Representation of Linear Transformations
Last Updated: June 22, 2024Addition 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...
