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let $T$ be a linear transformation from $\mathbb{R}^2 \to \mathbb{R}^3$ and $T(-1,0)=(2,1,1)$ and $T(1,1)=(1,0,0)$. Then what is $T(x,y)$?

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2 Answers

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Since $T$ is a linear transformation, $T(-1,0)=-T(1,0)$.

$T(1,0) = (-2,-1,-1)$.

$T(0,1) = T(1,1)-T(1,0) = (3,1,1)$.

So $T(x,y) = xT(1,0)+yT(0,1) = (-2x+3y,-x+y,-x+y)$

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Let $(x,y) \in \mathbb R^2$. Determine $t,s$ such that

$(x,y)=s(-1,0)+t(1,1)$.

Then: $T(x,y)=s(2,1,1)+s(1,0,0)$.

Can you proceed ?

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