Tutorial note 5

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Some interesting application of dimension: We have mentioned that every linear transformation $T:\R^n\to \R^m$ with $n>m$ cannot be injective in tutorial note 3. With the concept of dimension, it is very simple!

 Fact. Let the linear transformation $S:V\to W$ be injective. If $\{v_1,v_2,\dots,v_n\}$ is linearly independent, so is $\{Sv_1,Sv_2,\dots,Sv_n\}$.

This is the key fact to explain why $T$ cannot be injective. Suppose $T$ is injective, let $\{e_1,\dots,e_n\}$ be a standard basis of $\R^n$, then \[
\{Te_1,\dots,Te_n\}\subseteq \R^m
\] is also linearly independent, hence \[
n=\dim (\spann \{Te_1,\dots,Te_n\})\leq \dim \R^m=m,
\]but $m<n$ by definition, a contradiction.