| course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
38 |
|
Paper 1, Section I, F |
2012 |
Define the notions of basis and dimension of a vector space. Prove that two finitedimensional real vector spaces with the same dimension are isomorphic.
In each case below, determine whether the set $S$ is a basis of the real vector space $V:$
(i) $V=\mathbb{C}$ is the complex numbers; $S={1, i}$.
(ii) $V=\mathbb{R}[x]$ is the vector space of all polynomials in $x$ with real coefficients; $S={1,(x-1),(x-1)(x-2),(x-1)(x-2)(x-3), \ldots} .$
(iii) $V={f:[0,1] \rightarrow \mathbb{R}} ; S=\left{\chi_{p} \mid p \in[0,1]\right}$, where
$$\chi_{p}(x)= \begin{cases}1 & x=p \ 0 & x \neq p\end{cases}$$