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Consider this matrix transformation: [[-1,-1,2],[-3,1,2],[-1,2,3]] What is the image of [[-1],[-1],[3]] under this transformation?

Consider this matrix transformation:\newline[−1amp;−1amp;2−3amp;1amp;2−1amp;2amp;3] \left[\begin{array}{ccc} -1 & -1 & 2 \\ -3 & 1 & 2 \\ -1 & 2 & 3 \end{array}\right] \newlineWhat is the image of [−1−13] \left[\begin{array}{c}-1 \\ -1 \\ 3\end{array}\right] under this transformation?

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Full solution

Q. Consider this matrix transformation:\newline[−1−12−312−123] \left[\begin{array}{ccc} -1 & -1 & 2 \\ -3 & 1 & 2 \\ -1 & 2 & 3 \end{array}\right] \newlineWhat is the image of [−1−13] \left[\begin{array}{c}-1 \\ -1 \\ 3\end{array}\right] under this transformation?
  1. Write Matrix and Vector: Write down the matrix and the vector to be transformed.\newlineMatrix AA:\newline[−1amp;−1amp;2 −3amp;1amp;2 −1amp;2amp;3]\begin{bmatrix}-1 & -1 & 2 \ -3 & 1 & 2 \ -1 & 2 & 3\end{bmatrix}\newlineVector vv:\newline[−1 −1 3]\begin{bmatrix}-1 \ -1 \ 3\end{bmatrix}
  2. Perform Matrix Multiplication: Perform matrix multiplication to find the image of the vector under the transformation.\newlineTo multiply the matrix AA by the vector vv, we take the dot product of each row of AA with the vector vv.\newlineFirst entry (row 11 of AA with vv):\newline(−1)⋅(−1)+(−1)⋅(−1)+(2)⋅(3)=1+1+6=8(-1)\cdot(-1) + (-1)\cdot(-1) + (2)\cdot(3) = 1 + 1 + 6 = 8\newlineSecond entry (row 22 of AA with vv):\newline(−3)⋅(−1)+(1)⋅(−1)+(2)⋅(3)=3−1+6=8(-3)\cdot(-1) + (1)\cdot(-1) + (2)\cdot(3) = 3 - 1 + 6 = 8\newlineThird entry (row 33 of AA with vv):\newlinevv22
  3. Write Resulting Image Vector: Write down the resulting image vector.\newlineThe image of the vector vv under the transformation by matrix AA is:\newline\left[\begin{array}{c}\(\newline8 (\newline\)8 (\newline\)8\newline\end{array}\right]\)

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