# Given a linear transformation and basis, verify $[T]_{\beta}^{\alpha}[v]_{\alpha} = T[(v)]_{\beta}$ . (More details in description) [closed]

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 $$T[(v)]_{\beta}$$?

$$[v]_\alpha$$ denotes the column vector consisting of scalars of $$v$$ w.r.t the ordered basis $$\alpha$$ and similar for other.
• No, if $v=(1,2,0)=0(1,1,1)+2(1,1,0)-1(1,0,0) \text{then} [v]_{\alpha}=(0,2,-1)^t$ – Dey May 9 at 16:07