14-240/Tutorial-November11

DBN: Classes: 2014-15: 14-240 / Navigation Welcome to Math 240! (additions to this web site no longer count towards good deed points) # Week of... Notes and Links 1 Sep 8 About This Class, What is this class about? (PDF, HTML), Monday, Wednesday 2 Sep 15 HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf 3 Sep 22 HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf 4 Sep 29 HW3, Wednesday, Tutorial, HW3_solutions.pdf 5 Oct 6 HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf 6 Oct 13 No Monday class (Thanksgiving), Wednesday, Tutorial 7 Oct 20 HW5, Term Test at tutorials on Tuesday, Wednesday 8 Oct 27 HW6, Monday, Why LinAlg?, Wednesday, Tutorial 9 Nov 3 Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial 10 Nov 10 HW8, Monday, Tutorial 11 Nov 17 Monday-Tuesday is UofT November break 12 Nov 24 HW9 13 Dec 1 Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial F Dec 8 The Final Exam Register of Good Deeds Add your name / see who's in!

Boris

Background

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} be a finite dimensional vector space over a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , $B=\{v_{1},v_{2},v_{3},...,v_{n}\}$ be an ordered basis of $V$ and $v\in V$. Then $v=\displaystyle \sum _{i=1}^{n}c_{i}v_{i}$ where $c_{i}\in F$. Then the coordinate vector of $v$ relative to $B$ is the column vector ${\begin{pmatrix}c_{1}\\c_{2}\\c_{3}\\...\\c_{n}\end{pmatrix}}$.

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

Boris's Problems

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

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

Q2. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T:P_n \to F} be a linear transformation that is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(f(x)) = f(0)} . What is the matrix representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B, K} ?