# 14-240/Tutorial-November11

## Boris

#### Useful Definitions

Let ${\displaystyle V}$ be a finite dimensional vector space over a field ${\displaystyle F}$, ${\displaystyle B=\{v_{1},v_{2},v_{3},...,v_{n}\}}$ be an ordered basis of ${\displaystyle V}$ and ${\displaystyle v\in V}$. Then ${\displaystyle v=\displaystyle \sum _{i=1}^{n}c_{i}v_{i}}$ where ${\displaystyle c_{i}\in F}$. Then the coordinate vector of ${\displaystyle v}$ relative to ${\displaystyle B}$ is the column vector ${\displaystyle {\begin{pmatrix}c_{1}\\c_{2}\\c_{3}\\...\\c_{n}\end{pmatrix}}}$.

Let ${\displaystyle W}$ be a finite dimensional vector space over the same field ${\displaystyle F}$ and ${\displaystyle K=\{v_{1},v_{2},v_{3},...,v_{m}\}}$ be an ordered basis of ${\displaystyle W}$. Define a linear transformation ${\displaystyle T:V\to W}$. Then ${\displaystyle T(v_{j})=\displaystyle \sum _{i=1}^{m}c_{ij}T(v_{j})}$ where ${\displaystyle c_{ij}\in F}$. Then the matrix representation of ${\displaystyle T}$ in the ordered bases ${\displaystyle B,K}$ is the matrix ${\displaystyle {\begin{pmatrix}c_{11}&c_{12}&c_{13}&...&c_{1n}\\c_{21}&c_{22}&c_{23}&...&c_{2n}\\c_{31}&c_{32}&c_{33}&...&c_{3n}\\c_{21}&c_{22}&c_{23}&...&c_{2n}\\...&...&...&...&...\\c_{m1}&c_{m2}&c_{m3}&...&c_{mn}\end{pmatrix}}}$.

#### Boris's Problems

Let ${\displaystyle B}$ be the standard ordered basis of ${\displaystyle P_{n}(F)}$ and ${\displaystyle K}$ be the standard ordered basis of ${\displaystyle F}$.

Q1. What is the coordinate vector of ${\displaystyle x^{2}+x^{5}}$ relative to ${\displaystyle B}$?

Q2. Let ${\displaystyle T:P_{n}\to F}$ be a linear transformation that is defined by ${\displaystyle T(f(x))=f(0)}$. What is the matrix representation of ${\displaystyle T}$ in ${\displaystyle B,K}$?