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Question 4.1.5: Define a mapping T: R²→ R² by T([ x y ])= [e^x e^y ] Determi...

Define a mapping T: R²→ R² by

T\left(\begin{bmatrix} x \\ y \end{bmatrix} \right) = \left(\begin{bmatrix} e^{x} \\ e^{y} \end{bmatrix} \right)

Determine whether T is a linear transformation.

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Since

T (0) = T\left(\begin{bmatrix} 0 \\ 0 \end{bmatrix} \right) = \left(\begin{bmatrix} e^{0} \\ e^{0} \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}

by the contrapositive of Proposition 1, we know that T is not a linear transformation.

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