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

Consider this matrix transformation:\newline[2amp;3amp;−3amp;−1] \left[\begin{array}{cc} 2 & 3 \\ & \\ -3 & -1 \end{array}\right] \newlineWhat is the image of [41] \left[\begin{array}{l}4 \\ 1\end{array}\right] under this transformation?

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Full solution

Q. Consider this matrix transformation:\newline[23−3−1] \left[\begin{array}{cc} 2 & 3 \\ & \\ -3 & -1 \end{array}\right] \newlineWhat is the image of [41] \left[\begin{array}{l}4 \\ 1\end{array}\right] under this transformation?
  1. Identify Matrix and Vector: Identify the matrix and the vector to be transformed. Matrix A=[2amp;3 −3amp;−1]A = \left[\begin{array}{cc} 2 & 3 \ -3 & -1 \end{array}\right] Vector v=[4 1]v = \left[\begin{array}{c} 4 \ 1 \end{array}\right]
  2. Perform Matrix Multiplication: Perform matrix multiplication to find the image of the vector.\newlineImage of vv, Av = \begin{bmatrix}2 & 3\-3 & -1\end{bmatrix} * \begin{bmatrix}4\1\end{bmatrix}\newlineTo multiply, take the dot product of the rows of AA with the columns of vv.
  3. Calculate First Element: Calculate the first element of the resulting vector.\newlineFirst element = (2×4)+(3×1)=8+3=11(2 \times 4) + (3 \times 1) = 8 + 3 = 11
  4. Calculate Second Element: Calculate the second element of the resulting vector.\newlineSecond element = (−3×4)+(−1×1)=−12−1=−13(-3 \times 4) + (-1 \times 1) = -12 - 1 = -13
  5. Combine Results: Combine the results to get the image of the vector.\newlineThe image of \begin{bmatrix}4\1\end{bmatrix} under the transformation is \begin{bmatrix}11\-13\end{bmatrix}.

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