14-240/Tutorial-November11

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Contents

Boris

Useful Definitions

Let V be a finite dimensional vector space over a field 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_iv_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 T:P_n \to F be a linear transformation that is defined by T(f(x)) = f(0). What is the matrix representation of T in B, K?

Nikita