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**Isomorphism** 1)

$\displaystyle T \begin{bmatrix}x_{1}\\x_{2}\end{bmatrix} = det \begin{bmatrix}x_{1}&1\\x_{2}&1\end{bmatrix} = x_1 - x_2 = \begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}x_{1}\\x_{2}\end{bmat rix}$

T is indeed linear!

Ker T = $\displaystyle \{{\bf x} = (x_1,x_2) \in \mathbb{R}^2| T{\bf x} = x_1 - x_2 = 0\}$

Ker T is the line passing through origin with a slope of $\displaystyle 45^{\circ}$ with the $\displaystyle x_2$ (or $\displaystyle x_1$) axis on the $\displaystyle x_1-x_2$ plane.