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Let $T : R^3 -> R^2 $ be the linear transformation defined by $T(x,y,z) = (3x +2y -4z, x-5y +3z)$, and let $\alpha$ = {(1,1,1), (1,1,0), (1,0,0)} and $\beta$ = {(1,3),(2,5)}.
$Verify$ $ [T]_{\beta}^{\alpha}[v]_{\alpha} = T[(v)]_{\beta}$

This is a sub question of my assignment and I did not understand the verification part. How do you define $[v]_{\alpha}$ and $ T[(v)]_{\beta}$?

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$[v]_\alpha$ denotes the column vector consisting of scalars of $v$ w.r.t the ordered basis $\alpha$ and similar for other.

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  • $\begingroup$ Thank you. Could you give me a small example if v = (1,2,0)? Would [v]α =1*(1,1,1)+2*(1,1,0) + 0*(1,0,0)? $\endgroup$ – Navi May 9 at 15:30
  • $\begingroup$ No, if $v=(1,2,0)=0(1,1,1)+2(1,1,0)-1(1,0,0) \text{then} [v]_{\alpha}=(0,2,-1)^t$ $\endgroup$ – Dey May 9 at 16:07
  • $\begingroup$ Got it. Thank you so much! $\endgroup$ – Navi May 9 at 16:30

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