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

Consider this matrix transformation:\newline[−4−223] \left[\begin{array}{cc} -4 & -2 \\ 2 & 3 \end{array}\right] \newlineWhat is the image of [3−1] \left[\begin{array}{c}3 \\ -1\end{array}\right] under this transformation?

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Full solution

Q. Consider this matrix transformation:\newline[−4−223] \left[\begin{array}{cc} -4 & -2 \\ 2 & 3 \end{array}\right] \newlineWhat is the image of [3−1] \left[\begin{array}{c}3 \\ -1\end{array}\right] under this transformation?
  1. Perform Matrix Multiplication: To find the image of the vector \left[\begin{array}{c}3\-1\end{array}\right] under the transformation defined by the matrix \left[\begin{array}{cc}-4 & -2\2 & 3\end{array}\right], we need to perform matrix multiplication.
  2. Calculate Dot Products: The multiplication process involves 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 First Element: Calculating the first element of the resulting vector: (−4×3)+(−2×−1)=−12+2=−10(-4 \times 3) + (-2 \times -1) = -12 + 2 = -10.
  4. Calculate Second Element: Calculating the second element of the resulting vector: (2×3)+(3×−1)=6−3=3(2 \times 3) + (3 \times -1) = 6 - 3 = 3.
  5. Combine Results: Combining the results from the previous steps, the image of the vector [3 −1]\begin{bmatrix} 3 \ -1 \end{bmatrix} under the given transformation is [−10 3]\begin{bmatrix} -10 \ 3 \end{bmatrix}.

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