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Consider this matrix transformation: [[-2,4],[,],[3,-1]] What is the image of [[-1],[-2]] under this transformation?

Consider this matrix transformation:\newline[−243−1] \left[\begin{array}{cc} -2 & 4 \\ & \\ 3 & -1 \end{array}\right] \newlineWhat is the image of [−1−2] \left[\begin{array}{l}-1 \\ -2\end{array}\right] under this transformation?

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Q. Consider this matrix transformation:\newline[−243−1] \left[\begin{array}{cc} -2 & 4 \\ & \\ 3 & -1 \end{array}\right] \newlineWhat is the image of [−1−2] \left[\begin{array}{l}-1 \\ -2\end{array}\right] under this transformation?
  1. Multiply Matrix by Vector: To find the image of the vector \begin{bmatrix}-1\-2\end{bmatrix} under the given transformation, we need to multiply the matrix \begin{bmatrix}-2 & 4\3 & -1\end{bmatrix} by the vector \begin{bmatrix}-1\-2\end{bmatrix}.
  2. Calculate First Element: The multiplication of a 2×22 \times 2 matrix with a 2×12 \times 1 vector is done by taking the dot product of the rows of the matrix with the columns of the vector. The first element of the resulting vector is the dot product of the first row of the matrix with the vector, and the second element is the dot product of the second row of the matrix with the vector.
  3. Calculate Second Element: The first element of the resulting vector is calculated as follows:\newline(−2)×(−1)+4×(−2)=2−8=−6(-2) \times (-1) + 4 \times (-2) = 2 - 8 = -6.
  4. Find Image of Vector: The second element of the resulting vector is calculated as follows:\newline3×(−1)+(−1)×(−2)=−3+2=−13 \times (-1) + (-1) \times (-2) = -3 + 2 = -1.
  5. Find Image of Vector: The second element of the resulting vector is calculated as follows: 3×(−1)+(−1)×(−2)=−3+2=−13 \times (-1) + (-1) \times (-2) = -3 + 2 = -1. Therefore, the image of the vector [[−1],[−2]][[-1],[-2]] under the given transformation is the vector [[−6],[−1]][[-6],[-1]].

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