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    Linearity tests for maps $\RR^2\to\RR^2$

    E-HK-31-1

    Area
    Algebra
    Topics
    Linear Algebra, Linear Transformations
    Status
    draft

    Which of the following functions \(T\) from \(\RR^2\) into \(\RR^2\) are linear transformations?

    1. \(T(x_1, x_2) = (1 + x_1, x_2)\);

    2. \(T(x_1, x_2) = (x_2, x_1)\);

    3. \(T(x_1, x_2) = (x_1^2, x_2)\);

    4. \(T(x_1, x_2) = (\sin x_1, x_2)\);

    5. \(T(x_1, x_2) = (x_1 - x_2, 0)\).

    Solution

    Exactly (b) and (e) are linear.

    1. is not linear because \(T(0,0)=(1,0)\ne0\).

    2. is linear since \[ T(x_1,x_2)=(x_2,x_1) =\begin{bmatrix}0&1\\1&0\end{bmatrix} \begin{bmatrix}x_1\\x_2\end{bmatrix}. \]

    3. is not linear because, for example, \[ T(2,0)=(4,0)\ne2(1,0)=2T(1,0). \]

    4. is not linear since \[ T(2x_1,0)=(\sin 2x_1,0) \] is not generally equal to \(2(\sin x_1,0)\).

    5. is linear because \[ T(x_1,x_2)=(x_1-x_2,0) =\begin{bmatrix}1&-1\\0&0\end{bmatrix} \begin{bmatrix}x_1\\x_2\end{bmatrix}. \]

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