Finding the image of an arbitrary vector.

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$v_1$ and $v_2$ are linearly independent and form a basis. So you only need to write an arbitrary vector as linear combination of them.

Let $$\begin{bmatrix} x \\ y \end{bmatrix}=a\begin{bmatrix} 1 \\ 1 \end{bmatrix}+b\begin{bmatrix}2 \\ 3\end{bmatrix}$$ or $$\begin{bmatrix} 1 & 2 \\ 1 & 3\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}x\\y\end{bmatrix}$$ $$\begin{bmatrix} a \\ b \end{bmatrix}=\begin{bmatrix}1&2\\1&3\end{bmatrix}^{-1}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3&-2\\-1 & 1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix} 3x-2y\\-x+y \end{bmatrix}$$ So $$T\begin{bmatrix} x\\y\end{bmatrix}=T\left( (3x-2y)v_1+(-x+y)v_2\right)$$ $$=(3x-2y)T(v_1)+(-x+y)T(v_2)$$ $$=(3x-2y)\begin{bmatrix}-11\\4\end{bmatrix}+(-x+y)\begin{bmatrix}-28\\13\end{bmatrix}$$ $$=\begin{bmatrix}-5x-6y\\-x+5y\end{bmatrix}$$

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Shammy
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Shammy

Software Engineer

Updated on August 15, 2022

Comments

  • Shammy
    Shammy about 1 year

    I'm not exactly sure how to find the image of an arbitrary vector

    enter image description here